Research ArticleA Theoretical Model for the Transmission Dynamics ofthe Buruli Ulcer with Saturated Treatment

Ebenezer Bonyah1 Isaac Dontwi1 and Farai Nyabadza2

1 Department of Mathematics Kwame Nkrumah University of Science and Technology Kumasi Ghana2Department of Mathematical Science University of Stellenbosch Private Bag X1 Matieland 7602 South Africa

Correspondence should be addressed to Farai Nyabadza fnyabagmailcom

Received 16 May 2014 Revised 5 August 2014 Accepted 6 August 2014 Published 21 August 2014

Academic Editor Chung-Min Liao

Copyright copy 2014 Ebenezer Bonyah et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The management of the Buruli ulcer (BU) in Africa is often accompanied by limited resources delays in treatment and macilentcapacity in medical facilitiesThese challenges limit the number of infected individuals that access medical facilities While most ofthemathematicalmodelswith treatment assume a treatment function proportional to the number of infected individuals in settingswith such limitations this assumptionmay not be valid To capture these challenges a mathematical model of the Buruli ulcer witha saturated treatment function is developed and studiedThemodel is a coupled system of two submodels for the human populationand the environmentWe examine the stability of the submodels and carry out numerical simulationsThemodel analysis is carriedout in terms of the reproduction number of the submodel of environmental dynamics The dynamics of the human populationsubmodel are found to occur at the steady states of the submodel of environmental dynamics Sensitivity analysis is carried out onthe model parameters and it is observed that the BU epidemic is driven by the dynamics of the environment The model suggeststhat more effort should be focused on environmental management The paper is concluded by discussing the public implicationsof the results

1 Introduction

TheBuruli ulcer disease (BU) is a rapidly emerging neglectedtropical disease caused byMycobacterium ulcerans (M ulcer-ans) [1 2] It is a poorly understood disease that is associatedwith rapid environmental changes to the landscapes suchas deforestation construction and mining [3ndash6] It is aserious necrotizing cutaneous infection which can result incontracture deformities and amputations of the affected limb[3 7] Very little is known about the ecology of theMulceransin the environment and their distribution patterns [3] Thesurvival of vectors or pathogens in the environment can bedirectly or indirectly influenced by landscape features such asland use and cover types These features influence the vectoror pathogenrsquos ability to survive in the environment or to betransmitted In most cases the dynamics of the reservoirsand vector depend on the management of the environmentResearch has shown that BU is highly prevalent in arsenic-enriched drainages and farmlands [8 9]

The lack of understanding of the dynamics of the inter-actions of humans the vectors and the BU transmissionprocesses severely hinders prevention and control programsHowever mathematical models have been used immenselyas tools for understanding the epidemiology of diseases andevaluating interventions They now play an important rolein policy making health-economic aspects emergency plan-ning and risk assessment control-programs evaluation andoptimizing various detection methods [10] The majority ofmathematicalmodels developed to date for disease epidemicsare compartmental Many of them assume that the transferrates between compartments are proportional to the individ-uals in a compartment In an environment where resourcesare limited and services lean this assumption is unrealisticIn particular the uptake rate of infected individuals intotreatment programs is often influenced by the capacity ofhealth care systems costs socioeconomic factors and theefficiency of health care services For BU the number ofpeople admitted for treatment is limited by the capacity

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 576039 14 pageshttpdxdoiorg1011552014576039

2 Computational and Mathematical Methods in Medicine

of health care services the cost of treatment distance tohospitals and health care facilities that are often few [11 12]BU treatment is by surgery and skin grafting or antibiotics Itis documented that antibiotics kill M ulcerans bacilli arrestthe disease and promote healing without relapse or reducethe extent of surgical excision [13] Improved treatmentoptions can alleviate the plight of sufferers These challengesall stem from the fact that many of the developing countrieshave limited resources

The demand for health care services often exceeds thecapacity of health care provision in cases where the infectedvisit modern medical facilities It will be thus plausible to usea saturated treatment function to model limited capacity inthe treatment of the BU see also [10 14 15]The transmissionof BU is driven by two processes firstly it occurs throughdirect contact withM ulcerans in the environment [1 16 17]and secondly it occurs through biting by water bugs [18 19]In this paper we capture these two modes of transmissionand also incorporate saturated treatmentThe aim is tomodeltheoretically the possible impact of the challenges associatedwith the treatment and management of the BU such asdelays in accessing treatment limited resources and fewmedical facilities to deal with the highly complex treatmentof the ulcer We also endeavour to holistically include themain forms of transmission of the disease in humans Thismakes the model richer than the few attempts made by someauthors see for instance [18]

This paper is arranged as follows In Section 2 we formu-late and establish the basic properties of themodelThemodelis analysed for stability in Section 3 Numerical simulationsare given in Section 4 In fact parameter estimation sensitiv-ity analysis and somenumerical results on the behavior of themodel are presented in this section The paper is concludedin Section 5

2 Model Formulation

21 Description The transmission dynamics of the BUinvolve three populations that of humans water bugs andthe M ulcerans Our model is thus a coupled system oftwo submodels The submodel of the human populationis an (119878

119867 119868119867 119879119867 119877119867) type model with 119878

119867denoting the

susceptible humans 119868119867

those infected with the BU 119879119867

those in treatment and 119877119867the recovered The total human

population is given by

119873119867= 119878119867+ 119868119867+ 119879119867+ 119877119867 (1)

The submodel of the water bugs and M ulcerans has threecompartments The population of water bugs is comprised ofsusceptiblewater bugs 119878

119882and the infectedwater bugs 119868

119882The

total water bugs population is given by

119873119882= 119878119882+ 119868119882 (2)

The third compartment 119863 is that of M ulcerans in theenvironment whose carrying capacity is 119870

119889 The possible

interrelations between humans the water bugs and envi-ronment are represented in Figure 1 As in [14 15] we alsoassume a saturation treatment function of the form

119891 (119868119867) =

120590119868119867

1 + 119868119867

(3)

where 120590 is the maximum treatment rate A different func-tion can however be chosen depending on the mod-elling assumptions The function that models the interactionbetween humans and M ulcerans has been used to modelcholera epidemics [20] and the references cited therein Wenote that if BU cases are few then 119891(119868

119867) asymp 120590119868

119867 which is

a linear function assumed in many compartmental modelsincorporating treatment see for instance [21 22] On theother hand if BU cases are many then 119891(119868

119867) asymp 120590 a constant

So for very large values of 119868119867of the uptake of BU patients into

treatment become constant thus reaching a saturation levelThe parameters 120573

1and 120573

2are the effective contact rates of

susceptible humanswith thewater bugs and the environmentrespectively Here 120573

1is the product of the biting frequency

of the water bugs on humans density of water bugs perhuman host and the probability that a bite will result in aninfection Also 120573

2is the product of density of M ulcerans

per human host and the probability that a contact will resultin an infection The parameter 119870

50gives the concentration

of M ulcerans in the environment that yield 50 chance ofinfection with BU

The dynamics of the susceptible population for whichnew susceptible populations enter at a rate of 120583

119867119873119867are given

by (4) Some BU sufferers do not recover with permanentimmunity they lose immunity at a rate 120579 and becomesusceptible againThe third termmodels the rate of infectionof susceptible populations and the last term describes thenatural mortality of the susceptible populations In thismodel the human population is assumed to be constant overthe modelling time with the birth and death rate (120583

119867) being

the same119889119878119867

119889119905= 120583119867119873119867+ 120579119877119867minus Λ119878119867minus 120583119867119878119867 (4)

whereΛ = 1205731119868119882119873119867+1205732119863(11987050+119863) and119891(119868

119867) is a function

that models saturation in the treatment of BUFor the population infected with the BU we have

119889119868119867

119889119905= Λ119878119867minus 119891 (119868

119867) minus 120583119867119868119867 (5)

Equation (5) depicts changes in the infected BU casesThe first term represents individuals who enter from thesusceptible pool driven by the force of infectionΛThe secondterm represents the treatment of BU cases modelled by thetreatment function119891(119868

119867)The last term represents the natural

mortality of infected humansEquation (6)

119889119879119867

119889119905= 119891 (119868

119867) minus (120583

119867+ 120574) 119868119867 (6)

models the human BU cases under treatment In this regardthe first term represents themovement of BU cases into treat-ment and the second term with rates 120583

119867and 120574 respectively

represents natural mortality and recovery

Computational and Mathematical Methods in Medicine 3

D

SH IH THRH

SW IW

120579RH

120583HSH 120583HIH 120583HTH120583HRH

120583WNW 120583WIW

120583WSW

1205733SWD

120572IW

120583HNHf(IH)

Human population dynamics

Envi

ronm

enta

l dyn

amic

s

120583dD

ΛSH 120574TH

Figure 1 A schematic diagram for the model

For individuals who would have recovered from theinfection after treatment their dynamics are represented bythe following equation

119889119877119867

119889119905= 120574119868119867minus (120583119867+ 120579) 119877

119867 (7)

The first term denotes those who recover at a per capita rate120574 and the second term with rates 120583

119867and 120579 respectively

represents the natural mortality and loss of immunityThe equations for the submodel of water bugs are

119889119878119882

119889119905= 120583119882119873119882minus 1205733

119878119882119863

119870119889

minus 120583119882119878119882 (8)

119889119868119882

119889119905= 1205733

119878119882119863

119870119889

minus 120583119882119868119882 (9)

Equation (8) tracks susceptible water bugs The first term isthe recruitment of water bugs at a rate of 120583119873

119882 The second

and third term model the infection rate of water bugs by Mulcerans at the rate of120573

3and the naturalmortality of thewater

bugs at a rate 120583119882 respectively Equation (9) deals with the

infectious class of the water bug population The first termsimply models the infection of water bugs and the secondterm models the clearance rate of infected water bugs 120583

119882

from the environmentThedynamics ofM ulcerans in the environment aremod-

elled by

119889119863

119889119905= 120572119868119882minus 120583119889

119863

119870119889

(10)

The first term models the shedding of M ulcerans byinfected water bugs into the environment and the second

term represents the removal ofM ulcerans from the environ-ment at the rate 120583

119889

System (4)ndash(10) is subject to the following initial condi-tions

119878119867 (0) = 1198781198670 gt 0 119868

119867 (0) = 1198681198670 gt 0

119879119867 (0) = 1198791198670 gt 0 119877

119867 (0) = 1198771198670 = 0

119878119882 (0) = 1198781198820 gt 0 119868

119882 (0) = 1198681198820 119863 (0) = 1198630 gt 0

(11)

It is easier to analyse themodels (4)ndash(10) in dimensionlessform Using the following substitutions

119904ℎ=119878119867

119873119867

119894ℎ=119868119867

119873119867

120591ℎ=119879119867

119873119867

119903ℎ=119877119867

119873119867

119904119908=119878119882

119873119882

119894119908=119868119882

119873119882

119909 =119863

119870119889

1198981=119873119882

119873119867

(12)

and given that 119904ℎ+ 119894ℎ+ 120591ℎ+ 119903ℎ= 1 119904

119908+ 119894119908= 1 and 0 le

119909 le 1 system (4)ndash(10) when decomposed into its subsystemsbecomes

119889119904ℎ

119889119905= (120583119867+ 120579) (1 minus 119904

ℎ) minus 120579 (119894

ℎ+ 120591ℎ) minus Λ119904

ℎ

119889119894ℎ

119889119905= Λ119904ℎminus

120590119894ℎ

1 + 119873119867119894ℎ

minus 120583119867119894ℎ

119889120591ℎ

119889119905=

120590119894ℎ

1 + 119873119867119894ℎ

minus (120583119867+ 120574) 120591

ℎ

(13)

4 Computational and Mathematical Methods in Medicine

119889119894119908

119889119905= 1205733(1 minus 119894119908) 119909 minus 120583

119882119894119908

119889119909

119889119905= 119894119908minus 120583119889119909

(14)

where = 120572119873119882119870119889 Λ = 120573

11198981(119873119882 119873119867)119894119908+ 1205732119909( + 119909)

and = 11987050119870119889 Given that the total number of bites made

by the water bugs must equal the number of bites received bythe humans119898

1(119873119882 119873119867) is a constant see [23]

3 Model Analysis

Ourmodel has two subsystems that are only coupled throughinfection term Our analysis will thus focus on the dynamicsof the environment first and then we consider how thesedynamics subsequently affect the human populationWe firstconsider the properties of the overall system before we lookat the decoupled system

31 Basic Properties Since the model monitors changes inthe populations of humans and water bugs and the densityofM ulcerans in the environment the model parameters andvariables are nonnegativeThe biologically feasible region forthe systems (13)-(14) is in R5

+and is represented by the set

Γ = (119904ℎ 119894ℎ 120591ℎ 119894119908 119909) isin R

5

+| 0 le 119904

ℎ+ 119894ℎ+ 120591ℎle 1

0 le 119894119908le 1 0 le 119909 le

120583119889

(15)

where the basic properties of local existence uniquenessand continuity of solutions are valid for the Lipschitziansystems (13)-(14)Thepopulations described in thismodel areassumed to be constant over the modelling time

We can easily establish the positive invariance of Γ Giventhat 119889119909119889119905 = 119894

119908minus 120583119889119909 le minus 120583

119889119909 we have 119909 le 120583

119889 The

solutions of systems (13)-(14) starting in Γ remain in Γ for all119905 gt 0 The 120596-limit sets of systems (13)-(14) are contained inΓ It thus suffices to consider the dynamics of our system inΓ where the model is epidemiologically and mathematicallywell posed

32 Positivity of Solutions For any nonnegative initial condi-tions of systems (13)-(14) the solutions remain nonnegativefor all 119905 isin [0infin) Here we prove that all the stated variablesremain nonnegative and the solutions of the systems (13)-(14)with nonnegative initial conditionswill remain positive for all119905 gt 0 We have the following proposition

Proposition 1 For positive initial conditions of systems (13)-(14) the solutions 119904

ℎ(119905) 119894ℎ(119905) 120591ℎ(119905) 119894119908(119905) and 119909(119905) are non-

negative for all 119905 gt 0

Proof Assume that

= sup 119905 gt 0 119904ℎgt 0 119894ℎgt 0 120591ℎgt 0 119894119908gt 0 119909 gt 0 isin (0 119905]

(16)

Thus gt 0 and it follows directly from the first equation ofthe subsystem (13) that

119889119904ℎ

119889119905le (120583119867+ 120579) minus [(120583

119867+ 120579) + Λ] 119904

ℎ (17)

This is a first order differential equation that can easily besolved using an integrating factor For a nonconstant force ofinfection Λ we have

119904ℎ() le 119904

ℎ (0) exp[minus((120583119867 + 120579) + int

0

Λ (119904) 119889119904)]

+ exp[minus((120583119867+ 120579) + int

0

Λ (119904) 119889119904)]

times [int

0

(120583119867+ 120579) 119890

((120583119867+120579)+int

0Λ(119897)119889119897)

119889]

(18)

Since the right-hand side of (18) is always positive thesolution 119904

ℎ(119905) will always be positive If Λ is constant this

result still holdsFrom the second equation of subsystem (13)

119889119894ℎ

119889119905ge minus (120583

119867+ 120590) 119894ℎge 119894ℎ (0) exp [minus (120583119867 + 120590) 119905] gt 0 (19)

The third equation of subsystem (13) yields

119889120591ℎ

119889119905ge minus (120583

119867+ 120574) 120591

ℎge 120591ℎ (0) exp [minus (120583119867 + 120574) 119905] gt 0

(20)

Similarly we can show that 119894119908(119905) gt 0 and 119909(119905) gt 0 for all 119905 gt 0

and this completes the proof

33 Environmental Dynamics The subsystem (14) representsthe dynamics of water bugs and M ulcerans in the environ-ment From the second equation we have

119909lowast=119894lowast

119908

120583119889

119894lowast

119908= 0 or 119894

lowast

119908= 1 minus

1

R119879

(21)

where

R119879=

1205733

120583119889120583119882

(22)

In this case 119909lowast = (120583119889)(1 minus (1R

119879))

The case 119894lowast119908= 0 yields the infection free equilibrium point

of the environmental dynamics submodel given by

E0= (0 0) (23)

The submodel also has an endemic equilibrium given by

E1= (120583

119882(R119879minus 1) 120583

119889120583119882(R119879minus 1)) (24)

Remark 2 It is important to note that the R119879is our model

reproduction number for the BU epidemic in the presenceof treatment driven by the dynamics of the water bug and

Computational and Mathematical Methods in Medicine 5

M ulcerans in the environment A reproduction numberusually defined as the average of the number of secondarycases generated by an index case in a naive population isa key threshold parameter that determines whether the BUdisease persists or vanishes in the population In this case itrepresents the number of secondary cases of infected waterbugs generated by the shedded M ulcerans in the environ-ment R

119879determines the infection in the environment and

subsequently in the human population We can alternativelyuse the next generation operator method [24 25] to derivethe reproduction number A similar valuewas obtained undera square root sign in this case The reproduction number isindependent of the parameters of the human population evenwhen the two submodels are combined It depends on the lifespans of the water bugs andM ulcerans in the environmentthe shedding and infection rates of the water bugs So theinfection is driven by the water bug population and thedensity of the bacterium in the environment The modelreproduction number increases linearly with the sheddingrate of theM ulcerans into the environment and the effectivecontact rate between the water bugs and M ulcerans Thisimplies that the control and management of the ulcer largelydepend on environmental management

331 Stability of E0

Theorem 3 The infection free equilibrium E0is globally stable

whenR119879lt 1 and unstable otherwise

Proof We propose a Lyapunov function of the form

V (119905) = 119894119908 +1205733

120583119889

119909 (25)

The time derivative of (25) is

V =119889119894119908

119889119905+1205733

120583119889

119889119909

119889119905

le 120583119882(R119879minus 1) 119894119908

(26)

WhenR119879le 1 V is negative and semidefinite with equality

at the infection free equilibrium andor at R119879

= 1 Sothe largest compact invariant set in Γ such that V119889119905 le 0

when R119879le 1 is the singleton E

0 Therefore by the LaSalle

Invariance Principle [26] the infection free equilibriumpointE0is globally asymptotically stable if R

119879lt 1 and unstable

otherwise

332 Stability of E1

Theorem 4 The endemic steady state E1of the subsystem (14)

is locally asymptotically stable ifR119879gt 1

Proof The Jacobian matrix of system (14) at the equilibriumpoint E

1is given by

119869E1 = (minus120583119882

1205733

minus120583119889

) (27)

Given that the trace of 119869E1 is negative and the determinant isnegative if R

119879gt 1 we can thus conclude that the unique

endemic equilibrium is locally asymptotically stable when-ever R

119879gt 1

Theorem 5 If R119879gt 1 then the unique endemic equilibrium

E1is globally stable in the interior of Γ

Proof We now prove the global stability of endemic steadystate E

1whenever it exists using the Dulac criterion and the

Poincare-Bendixson theorem The proof entails the fact thatwe begin by ruling out the existence of periodic orbits in Γusing the Dulac criteria [27] Defining the right-hand side of(14) by (119865(119894

119908 119909) 119866(119894

119908 119909)) we can construct a Dulac function

B (119894119908 119909) =

1

1205733119894119908119909 119894119908gt 0 119909 gt 0 (28)

We will thus have

120597 (119865B)

120597119894119908

+120597 (119866B)

120597119909= minus(

1

1198942119908

+

12057331199092) lt 0 (29)

Thus subsystem (14) does not have a limit cycle in Γ FromTheorem 4 ifR

119879gt 1 thenE

1is locally asymptotically stable

A simple application of the classical Poincare-Bendixsontheorem and the fact that Γ is positively invariant sufficeto show that the unique endemic steady state is globallyasymptotically stable in Γ

34 Dynamics of BU in the Human Population Our ultimateinterest is to determine how the dynamics of water bugs andM ulcerans impact the human population The overall goalis to mitigate the influence of the M ulcerans on the humanpopulation We can actually evaluate the force of infection sothat

Λ = (R119879minus 1) 120583

119882(11989811205731120583119889+

1205732

+ (R119879minus 1) 120583

119882

)

(30)

This means that the analysis of submodel (13) is subject toR119879gt 1 Our force of infection is thus now a function of

the reproduction number of submodel (14) and is constantfor any given value of the reproduction number Figure 2 is aplot of Λ versusR

119879

Using the second equation of system (13) we can evaluate119904lowast

ℎso that

119904lowast

ℎ= ([120590119894

lowast

ℎ+ 120583ℎ119894ℎ(1 + 119873

119867119894lowast

ℎ)] [ + 120583

119882(R119879minus 1)])

times ((1 + 119873119867119894lowast

ℎ) [11989811205731120583119889120583119882(R119879minus 1)

times + 120583119882(R119879minus 1)

+1205732120583119882(R119879minus 1)] )

minus1

(31)

From the third equation of (13) we have

120591lowast

ℎ=

120590119894lowastℎ

(1 + 119873119867119894lowastℎ) (120574 + 120583

119867) (32)

6 Computational and Mathematical Methods in Medicine

00 05 10 15 20000

002

004

006

008

010

RT

Λ

Figure 2 The plot of the force of infection as a function ofR119879 The

force of infection increases linearly with the reproduction numberThe human population is at risk only ifR

119879gt 1

Substituting 119904lowastℎand 120591lowastℎin the first equation of (13) at the

steady state yields a quadratic equation in 119894lowastℎgiven by

119886119894lowast2

ℎ+ 119887119894lowast

ℎ+ 119888 = 0 (33)

where

119886 = 119873119867(120583119867+ 120574) (120583

119867+ 120579)

times (1205732120583119882(R119879minus 1) + ( + (R

119879minus 1) 120583

119882)

times [120583119867+ 11989811205731120583119889120583119882(R119879minus 1)] )

119887 = 120574120579120590 + 120583119867[120579120590 + 120574 (120579 + 120590)

+120583119867(120583119867+ 120574 + 120579 + 120590)]

+ (119877119901minus 1) [ (120583

119867+ 120574) (120583

119867+ 120579) (120583

119867+ 120590)

+ 1205732(120574120579 + (120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times(120583119867+ 120579) + 120583

119867(120583119867+ 120574 + 120579 + 120590))

+ 11989811205731120583119889120574120579 +(120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times (120583119867+ 120579) + 120583

119867

times (120583119867+ 120574 + 120579 + 120590)] 120583

119882

minus 1198981(119877119901minus 1)2

1205731120583119889(minus (120579120590 + 120574 (120579 + 120590))

+ 119873119867(120574 + 120583

119867) (120579 + 120583

119867)

minus120583119867(120574 + 120579 + 120590 + 120583

119867)) 1205832

119882

119888 = minus120583119882(120583119867+ 120574) (120583

119867+ 120579)

times [1205732+ 11989811205731120583119889( + 120583

119882(R119879minus 1))] (R

119879minus 1)

(34)

Clearly our model has two possible steady states given by

E119886

2= (119904lowast

ℎ 119894lowast+

ℎ 120591lowast

ℎ) E

119887

2= (119904lowast

ℎ 119894lowastminus

ℎ 120591lowast

ℎ) (35)

where 119894lowastplusmnℎ

are roots of the quadratic equation (33) We notethat if R

119879gt 1 we have 119886 gt 0 and 119888 lt 0 By Descartesrsquo rule

of signs irrespective of the sign of 119887 the quadratic equation(33) has one positive root the endemic equilibrium E119886

2= E2

We thus have the following result

Theorem 6 System (13) has a unique endemic equilibrium E2

wheneverR119879gt 1

Remark 7 It is important to note that when subsystem (14) isat its infection free steady state then the human populationwill also be free of the BUWe can easily establish the BU freeequilibrium in humans as Eℎ

0= (1 0 0) The existence of Eℎ

0

is thus subject to the water bugs and the environment beingfree ofM ulcerans

341 Local Stability of Eℎ0

Theorem 8 The disease free equilibrium Eℎ0whenever it exists

is locally asymptotically stable if R119879

lt 1 and unstableotherwise

Proof When R119879lt 1 then either there are no infections in

the water bugs or they are simply carriers So Eℎ0exists The

Jacobian matrix of system (13) at the disease free equilibriumpoint Eℎ

0is given by

119869Eℎ0

= (

minus (120583119867+ 120579) minus120579 minus120579

0 minus (120583119867+ 120590) 0

0 120590 minus (120583119867+ 120574)

) (36)

The eigenvalues of 119869Eℎ0

are 1205821= minus(120583

119867+120579) 120582

2= minus(120583

119867+120590) and

1205823= minus(120583

119867+ 120574) We can thus conclude that the disease free

equilibrium is locally asymptotically stable whenever R119879lt

1

342 Local Stability of E2

Theorem 9 The unique endemic equilibrium point E2is

locally asymptotically stable for R119879gt 1

Proof The Jacobian matrix at the endemic steady state E2is

given by

119869E2 =(

(

minus(120583119867+ 120579) minus Λ minus120579 minus120579

Λ minus120583119867minus

120590

(1 + 119894lowastℎ)2

0

0120590

(1 + 119894lowastℎ)2

minus (120583119867+ 120574)

)

)

(37)

Computational and Mathematical Methods in Medicine 7

If we let 120595 = 120590(1 + 119894lowastℎ)2 then the eigenvalues of 119869E2 are given

by the solutions of the characteristic polynomial

1205993+ 12057811205992+ 1205782120599 + 1205783= 0 (38)

where

1205781= (120583119867+ 120574) + (120583

119867+ 120579) + (120583

119867+ 120595) + Λ

1205782= (120583119867+ 120579) (120583

119867+ 120574Λ) + (120583

119867+ 120574) (120583

119867+ 120595 + Λ)

+ (120583119867+ 120579) (120583

119867+ 120595) + Λ120595

1205783= 120579Λ (120574 + 120595) + 120574 (120579 + Λ) 120595

+ 120583119867(Λ120595 + 120579 (Λ + 120595) + 120574 (120579 + Λ + 120595)

+120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

(39)

Using the Routh-Hurwitz criterion we note that 1205781gt

0 1205782gt 0 and 120578

3gt 0 The evaluation of 120578

11205782minus 1205783yields

(120579 + Λ) (120579 + 120595) (Λ + 120595) + 1205742(120579 + Λ + 120595) + 120574(120579 + Λ + 120595)

2

+ 2120583119867(1205742+ 1205792+ Λ2+ 3Λ120595 + 120595

2+ 3120579 (Λ + 120595)

+3120574 (120579 + Λ + 120595) + 4120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

gt 0

(40)

This establishes the necessary and sufficient conditions for allroots of the characteristic polynomial to lie on the left half ofthe complex plane So the endemic equilibrium E

2is locally

asymptotically stable

In the next section we establish the global stability of theendemic equilibrium using the approach according to Li andMuldowney [28] based on monotone dynamical systems andoutlined in Appendix A of [29 30]

343 Global Stability of the Endemic Equilibrium We beginby stating the following theorem

Theorem 10 If R119879gt 1 system (13) is uniformly persistent in

Γ the interior of ΓThe existence of Eℎ

0 only if R

119879gt 1 guarantees uniform

persistence [31] System (13) is said to be uniformly persistentif there exists a positive constant 119888 such that any solution(119904ℎ(119905) 119894ℎ(119905) 120591ℎ(119905)) with initial conditions (119904

ℎ(0) 119894ℎ(0) 120591ℎ(0)) isin

Γ satisfies

lim inf119905rarrinfin

119904ℎ (119905) gt 119888 lim inf

119905rarrinfin

119894ℎ (119905) gt 119888

lim inf119905rarrinfin

120591ℎ (119905) gt 119888

(41)

The proof of uniform persistence can be done usinguniform persistence results in [31 32]

Theorem 11 If Λ gt 120574 the endemic equilibrium point E2of

system (13) is globally asymptotically stable when R119879gt 1

Proof Using the arguments in [28] system (13) satisfiesassumptions 119867(1) and 119867(2) in Γ Let 119909 = (119904

ℎ 119894ℎ 120591ℎ) and

119891(119909) be the vector field of system (13) The Jacobian matrixcorresponding to system (13) is

119869(119904ℎ 119894ℎ 120591ℎ)

=((

(

minus(120579 + Λ + 120583119867) minus120579 minus120579

Λ minus(120590

(1 + 119873119867119894ℎ)2+ 120583119867) 0

0120590

(1 + 119873119867119894ℎ)2

minus (120583119867+ 120574)

))

)

(42)

The second additive compound matrix 119869[2](119904ℎ 119894ℎ 120591ℎ)

is given by

119869[2](119904ℎ 119894ℎ120591ℎ)

=

((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) 0

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

))))

)

(43)

We let the matrix function 119875 take the form

119875 (119904ℎ 119894ℎ 120591ℎ) = diag

119894ℎ

120591ℎ

119894ℎ

120591ℎ

119894ℎ

120591ℎ

(44)

We thus have

119875119891119875minus1= diag

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(45)

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

Computational and Mathematical Methods in Medicine 3

D

SH IH THRH

SW IW

120579RH

120583HSH 120583HIH 120583HTH120583HRH

120583WNW 120583WIW

120583WSW

1205733SWD

120572IW

120583HNHf(IH)

Human population dynamics

Envi

ronm

enta

l dyn

amic

s

120583dD

ΛSH 120574TH

Figure 1 A schematic diagram for the model

For individuals who would have recovered from theinfection after treatment their dynamics are represented bythe following equation

119889119877119867

119889119905= 120574119868119867minus (120583119867+ 120579) 119877

119867 (7)

The first term denotes those who recover at a per capita rate120574 and the second term with rates 120583

119867and 120579 respectively

represents the natural mortality and loss of immunityThe equations for the submodel of water bugs are

119889119878119882

119889119905= 120583119882119873119882minus 1205733

119878119882119863

119870119889

minus 120583119882119878119882 (8)

119889119868119882

119889119905= 1205733

119878119882119863

119870119889

minus 120583119882119868119882 (9)

Equation (8) tracks susceptible water bugs The first term isthe recruitment of water bugs at a rate of 120583119873

119882 The second

and third term model the infection rate of water bugs by Mulcerans at the rate of120573

3and the naturalmortality of thewater

bugs at a rate 120583119882 respectively Equation (9) deals with the

infectious class of the water bug population The first termsimply models the infection of water bugs and the secondterm models the clearance rate of infected water bugs 120583

119882

from the environmentThedynamics ofM ulcerans in the environment aremod-

elled by

119889119863

119889119905= 120572119868119882minus 120583119889

119863

119870119889

(10)

The first term models the shedding of M ulcerans byinfected water bugs into the environment and the second

term represents the removal ofM ulcerans from the environ-ment at the rate 120583

119889

System (4)ndash(10) is subject to the following initial condi-tions

119878119867 (0) = 1198781198670 gt 0 119868

119867 (0) = 1198681198670 gt 0

119879119867 (0) = 1198791198670 gt 0 119877

119867 (0) = 1198771198670 = 0

119878119882 (0) = 1198781198820 gt 0 119868

119882 (0) = 1198681198820 119863 (0) = 1198630 gt 0

(11)

It is easier to analyse themodels (4)ndash(10) in dimensionlessform Using the following substitutions

119904ℎ=119878119867

119873119867

119894ℎ=119868119867

119873119867

120591ℎ=119879119867

119873119867

119903ℎ=119877119867

119873119867

119904119908=119878119882

119873119882

119894119908=119868119882

119873119882

119909 =119863

119870119889

1198981=119873119882

119873119867

(12)

and given that 119904ℎ+ 119894ℎ+ 120591ℎ+ 119903ℎ= 1 119904

119908+ 119894119908= 1 and 0 le

119909 le 1 system (4)ndash(10) when decomposed into its subsystemsbecomes

119889119904ℎ

119889119905= (120583119867+ 120579) (1 minus 119904

ℎ) minus 120579 (119894

ℎ+ 120591ℎ) minus Λ119904

ℎ

119889119894ℎ

119889119905= Λ119904ℎminus

120590119894ℎ

1 + 119873119867119894ℎ

minus 120583119867119894ℎ

119889120591ℎ

119889119905=

120590119894ℎ

1 + 119873119867119894ℎ

minus (120583119867+ 120574) 120591

ℎ

(13)

4 Computational and Mathematical Methods in Medicine

119889119894119908

119889119905= 1205733(1 minus 119894119908) 119909 minus 120583

119882119894119908

119889119909

119889119905= 119894119908minus 120583119889119909

(14)

where = 120572119873119882119870119889 Λ = 120573

11198981(119873119882 119873119867)119894119908+ 1205732119909( + 119909)

and = 11987050119870119889 Given that the total number of bites made

by the water bugs must equal the number of bites received bythe humans119898

1(119873119882 119873119867) is a constant see [23]

3 Model Analysis

Ourmodel has two subsystems that are only coupled throughinfection term Our analysis will thus focus on the dynamicsof the environment first and then we consider how thesedynamics subsequently affect the human populationWe firstconsider the properties of the overall system before we lookat the decoupled system

31 Basic Properties Since the model monitors changes inthe populations of humans and water bugs and the densityofM ulcerans in the environment the model parameters andvariables are nonnegativeThe biologically feasible region forthe systems (13)-(14) is in R5

+and is represented by the set

Γ = (119904ℎ 119894ℎ 120591ℎ 119894119908 119909) isin R

5

+| 0 le 119904

ℎ+ 119894ℎ+ 120591ℎle 1

0 le 119894119908le 1 0 le 119909 le

120583119889

(15)

where the basic properties of local existence uniquenessand continuity of solutions are valid for the Lipschitziansystems (13)-(14)Thepopulations described in thismodel areassumed to be constant over the modelling time

We can easily establish the positive invariance of Γ Giventhat 119889119909119889119905 = 119894

119908minus 120583119889119909 le minus 120583

119889119909 we have 119909 le 120583

119889 The

solutions of systems (13)-(14) starting in Γ remain in Γ for all119905 gt 0 The 120596-limit sets of systems (13)-(14) are contained inΓ It thus suffices to consider the dynamics of our system inΓ where the model is epidemiologically and mathematicallywell posed

32 Positivity of Solutions For any nonnegative initial condi-tions of systems (13)-(14) the solutions remain nonnegativefor all 119905 isin [0infin) Here we prove that all the stated variablesremain nonnegative and the solutions of the systems (13)-(14)with nonnegative initial conditionswill remain positive for all119905 gt 0 We have the following proposition

Proposition 1 For positive initial conditions of systems (13)-(14) the solutions 119904

ℎ(119905) 119894ℎ(119905) 120591ℎ(119905) 119894119908(119905) and 119909(119905) are non-

negative for all 119905 gt 0

Proof Assume that

= sup 119905 gt 0 119904ℎgt 0 119894ℎgt 0 120591ℎgt 0 119894119908gt 0 119909 gt 0 isin (0 119905]

(16)

Thus gt 0 and it follows directly from the first equation ofthe subsystem (13) that

119889119904ℎ

119889119905le (120583119867+ 120579) minus [(120583

119867+ 120579) + Λ] 119904

ℎ (17)

This is a first order differential equation that can easily besolved using an integrating factor For a nonconstant force ofinfection Λ we have

119904ℎ() le 119904

ℎ (0) exp[minus((120583119867 + 120579) + int

0

Λ (119904) 119889119904)]

+ exp[minus((120583119867+ 120579) + int

0

Λ (119904) 119889119904)]

times [int

0

(120583119867+ 120579) 119890

((120583119867+120579)+int

0Λ(119897)119889119897)

119889]

(18)

Since the right-hand side of (18) is always positive thesolution 119904

ℎ(119905) will always be positive If Λ is constant this

result still holdsFrom the second equation of subsystem (13)

119889119894ℎ

119889119905ge minus (120583

119867+ 120590) 119894ℎge 119894ℎ (0) exp [minus (120583119867 + 120590) 119905] gt 0 (19)

The third equation of subsystem (13) yields

119889120591ℎ

119889119905ge minus (120583

119867+ 120574) 120591

ℎge 120591ℎ (0) exp [minus (120583119867 + 120574) 119905] gt 0

(20)

Similarly we can show that 119894119908(119905) gt 0 and 119909(119905) gt 0 for all 119905 gt 0

and this completes the proof

33 Environmental Dynamics The subsystem (14) representsthe dynamics of water bugs and M ulcerans in the environ-ment From the second equation we have

119909lowast=119894lowast

119908

120583119889

119894lowast

119908= 0 or 119894

lowast

119908= 1 minus

1

R119879

(21)

where

R119879=

1205733

120583119889120583119882

(22)

In this case 119909lowast = (120583119889)(1 minus (1R

119879))

The case 119894lowast119908= 0 yields the infection free equilibrium point

of the environmental dynamics submodel given by

E0= (0 0) (23)

The submodel also has an endemic equilibrium given by

E1= (120583

119882(R119879minus 1) 120583

119889120583119882(R119879minus 1)) (24)

Remark 2 It is important to note that the R119879is our model

reproduction number for the BU epidemic in the presenceof treatment driven by the dynamics of the water bug and

Computational and Mathematical Methods in Medicine 5

M ulcerans in the environment A reproduction numberusually defined as the average of the number of secondarycases generated by an index case in a naive population isa key threshold parameter that determines whether the BUdisease persists or vanishes in the population In this case itrepresents the number of secondary cases of infected waterbugs generated by the shedded M ulcerans in the environ-ment R

119879determines the infection in the environment and

subsequently in the human population We can alternativelyuse the next generation operator method [24 25] to derivethe reproduction number A similar valuewas obtained undera square root sign in this case The reproduction number isindependent of the parameters of the human population evenwhen the two submodels are combined It depends on the lifespans of the water bugs andM ulcerans in the environmentthe shedding and infection rates of the water bugs So theinfection is driven by the water bug population and thedensity of the bacterium in the environment The modelreproduction number increases linearly with the sheddingrate of theM ulcerans into the environment and the effectivecontact rate between the water bugs and M ulcerans Thisimplies that the control and management of the ulcer largelydepend on environmental management

331 Stability of E0

Theorem 3 The infection free equilibrium E0is globally stable

whenR119879lt 1 and unstable otherwise

Proof We propose a Lyapunov function of the form

V (119905) = 119894119908 +1205733

120583119889

119909 (25)

The time derivative of (25) is

V =119889119894119908

119889119905+1205733

120583119889

119889119909

119889119905

le 120583119882(R119879minus 1) 119894119908

(26)

WhenR119879le 1 V is negative and semidefinite with equality

at the infection free equilibrium andor at R119879

= 1 Sothe largest compact invariant set in Γ such that V119889119905 le 0

when R119879le 1 is the singleton E

0 Therefore by the LaSalle

Invariance Principle [26] the infection free equilibriumpointE0is globally asymptotically stable if R

119879lt 1 and unstable

otherwise

332 Stability of E1

Theorem 4 The endemic steady state E1of the subsystem (14)

is locally asymptotically stable ifR119879gt 1

Proof The Jacobian matrix of system (14) at the equilibriumpoint E

1is given by

119869E1 = (minus120583119882

1205733

minus120583119889

) (27)

Given that the trace of 119869E1 is negative and the determinant isnegative if R

119879gt 1 we can thus conclude that the unique

endemic equilibrium is locally asymptotically stable when-ever R

119879gt 1

Theorem 5 If R119879gt 1 then the unique endemic equilibrium

E1is globally stable in the interior of Γ

Proof We now prove the global stability of endemic steadystate E

1whenever it exists using the Dulac criterion and the

Poincare-Bendixson theorem The proof entails the fact thatwe begin by ruling out the existence of periodic orbits in Γusing the Dulac criteria [27] Defining the right-hand side of(14) by (119865(119894

119908 119909) 119866(119894

119908 119909)) we can construct a Dulac function

B (119894119908 119909) =

1

1205733119894119908119909 119894119908gt 0 119909 gt 0 (28)

We will thus have

120597 (119865B)

120597119894119908

+120597 (119866B)

120597119909= minus(

1

1198942119908

+

12057331199092) lt 0 (29)

Thus subsystem (14) does not have a limit cycle in Γ FromTheorem 4 ifR

119879gt 1 thenE

1is locally asymptotically stable

A simple application of the classical Poincare-Bendixsontheorem and the fact that Γ is positively invariant sufficeto show that the unique endemic steady state is globallyasymptotically stable in Γ

34 Dynamics of BU in the Human Population Our ultimateinterest is to determine how the dynamics of water bugs andM ulcerans impact the human population The overall goalis to mitigate the influence of the M ulcerans on the humanpopulation We can actually evaluate the force of infection sothat

Λ = (R119879minus 1) 120583

119882(11989811205731120583119889+

1205732

+ (R119879minus 1) 120583

119882

)

(30)

This means that the analysis of submodel (13) is subject toR119879gt 1 Our force of infection is thus now a function of

the reproduction number of submodel (14) and is constantfor any given value of the reproduction number Figure 2 is aplot of Λ versusR

119879

Using the second equation of system (13) we can evaluate119904lowast

ℎso that

119904lowast

ℎ= ([120590119894

lowast

ℎ+ 120583ℎ119894ℎ(1 + 119873

119867119894lowast

ℎ)] [ + 120583

119882(R119879minus 1)])

times ((1 + 119873119867119894lowast

ℎ) [11989811205731120583119889120583119882(R119879minus 1)

times + 120583119882(R119879minus 1)

+1205732120583119882(R119879minus 1)] )

minus1

(31)

From the third equation of (13) we have

120591lowast

ℎ=

120590119894lowastℎ

(1 + 119873119867119894lowastℎ) (120574 + 120583

119867) (32)

6 Computational and Mathematical Methods in Medicine

00 05 10 15 20000

002

004

006

008

010

RT

Λ

Figure 2 The plot of the force of infection as a function ofR119879 The

force of infection increases linearly with the reproduction numberThe human population is at risk only ifR

119879gt 1

Substituting 119904lowastℎand 120591lowastℎin the first equation of (13) at the

steady state yields a quadratic equation in 119894lowastℎgiven by

119886119894lowast2

ℎ+ 119887119894lowast

ℎ+ 119888 = 0 (33)

where

119886 = 119873119867(120583119867+ 120574) (120583

119867+ 120579)

times (1205732120583119882(R119879minus 1) + ( + (R

119879minus 1) 120583

119882)

times [120583119867+ 11989811205731120583119889120583119882(R119879minus 1)] )

119887 = 120574120579120590 + 120583119867[120579120590 + 120574 (120579 + 120590)

+120583119867(120583119867+ 120574 + 120579 + 120590)]

+ (119877119901minus 1) [ (120583

119867+ 120574) (120583

119867+ 120579) (120583

119867+ 120590)

+ 1205732(120574120579 + (120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times(120583119867+ 120579) + 120583

119867(120583119867+ 120574 + 120579 + 120590))

+ 11989811205731120583119889120574120579 +(120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times (120583119867+ 120579) + 120583

119867

times (120583119867+ 120574 + 120579 + 120590)] 120583

119882

minus 1198981(119877119901minus 1)2

1205731120583119889(minus (120579120590 + 120574 (120579 + 120590))

+ 119873119867(120574 + 120583

119867) (120579 + 120583

119867)

minus120583119867(120574 + 120579 + 120590 + 120583

119867)) 1205832

119882

119888 = minus120583119882(120583119867+ 120574) (120583

119867+ 120579)

times [1205732+ 11989811205731120583119889( + 120583

119882(R119879minus 1))] (R

119879minus 1)

(34)

Clearly our model has two possible steady states given by

E119886

2= (119904lowast

ℎ 119894lowast+

ℎ 120591lowast

ℎ) E

119887

2= (119904lowast

ℎ 119894lowastminus

ℎ 120591lowast

ℎ) (35)

where 119894lowastplusmnℎ

are roots of the quadratic equation (33) We notethat if R

119879gt 1 we have 119886 gt 0 and 119888 lt 0 By Descartesrsquo rule

of signs irrespective of the sign of 119887 the quadratic equation(33) has one positive root the endemic equilibrium E119886

2= E2

We thus have the following result

Theorem 6 System (13) has a unique endemic equilibrium E2

wheneverR119879gt 1

Remark 7 It is important to note that when subsystem (14) isat its infection free steady state then the human populationwill also be free of the BUWe can easily establish the BU freeequilibrium in humans as Eℎ

0= (1 0 0) The existence of Eℎ

0

is thus subject to the water bugs and the environment beingfree ofM ulcerans

341 Local Stability of Eℎ0

Theorem 8 The disease free equilibrium Eℎ0whenever it exists

is locally asymptotically stable if R119879

lt 1 and unstableotherwise

Proof When R119879lt 1 then either there are no infections in

the water bugs or they are simply carriers So Eℎ0exists The

Jacobian matrix of system (13) at the disease free equilibriumpoint Eℎ

0is given by

119869Eℎ0

= (

minus (120583119867+ 120579) minus120579 minus120579

0 minus (120583119867+ 120590) 0

0 120590 minus (120583119867+ 120574)

) (36)

The eigenvalues of 119869Eℎ0

are 1205821= minus(120583

119867+120579) 120582

2= minus(120583

119867+120590) and

1205823= minus(120583

119867+ 120574) We can thus conclude that the disease free

equilibrium is locally asymptotically stable whenever R119879lt

1

342 Local Stability of E2

Theorem 9 The unique endemic equilibrium point E2is

locally asymptotically stable for R119879gt 1

Proof The Jacobian matrix at the endemic steady state E2is

given by

119869E2 =(

(

minus(120583119867+ 120579) minus Λ minus120579 minus120579

Λ minus120583119867minus

120590

(1 + 119894lowastℎ)2

0

0120590

(1 + 119894lowastℎ)2

minus (120583119867+ 120574)

)

)

(37)

Computational and Mathematical Methods in Medicine 7

If we let 120595 = 120590(1 + 119894lowastℎ)2 then the eigenvalues of 119869E2 are given

by the solutions of the characteristic polynomial

1205993+ 12057811205992+ 1205782120599 + 1205783= 0 (38)

where

1205781= (120583119867+ 120574) + (120583

119867+ 120579) + (120583

119867+ 120595) + Λ

1205782= (120583119867+ 120579) (120583

119867+ 120574Λ) + (120583

119867+ 120574) (120583

119867+ 120595 + Λ)

+ (120583119867+ 120579) (120583

119867+ 120595) + Λ120595

1205783= 120579Λ (120574 + 120595) + 120574 (120579 + Λ) 120595

+ 120583119867(Λ120595 + 120579 (Λ + 120595) + 120574 (120579 + Λ + 120595)

+120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

(39)

Using the Routh-Hurwitz criterion we note that 1205781gt

0 1205782gt 0 and 120578

3gt 0 The evaluation of 120578

11205782minus 1205783yields

(120579 + Λ) (120579 + 120595) (Λ + 120595) + 1205742(120579 + Λ + 120595) + 120574(120579 + Λ + 120595)

2

+ 2120583119867(1205742+ 1205792+ Λ2+ 3Λ120595 + 120595

2+ 3120579 (Λ + 120595)

+3120574 (120579 + Λ + 120595) + 4120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

gt 0

(40)

This establishes the necessary and sufficient conditions for allroots of the characteristic polynomial to lie on the left half ofthe complex plane So the endemic equilibrium E

2is locally

asymptotically stable

In the next section we establish the global stability of theendemic equilibrium using the approach according to Li andMuldowney [28] based on monotone dynamical systems andoutlined in Appendix A of [29 30]

343 Global Stability of the Endemic Equilibrium We beginby stating the following theorem

Theorem 10 If R119879gt 1 system (13) is uniformly persistent in

Γ the interior of ΓThe existence of Eℎ

0 only if R

119879gt 1 guarantees uniform

persistence [31] System (13) is said to be uniformly persistentif there exists a positive constant 119888 such that any solution(119904ℎ(119905) 119894ℎ(119905) 120591ℎ(119905)) with initial conditions (119904

ℎ(0) 119894ℎ(0) 120591ℎ(0)) isin

Γ satisfies

lim inf119905rarrinfin

119904ℎ (119905) gt 119888 lim inf

119905rarrinfin

119894ℎ (119905) gt 119888

lim inf119905rarrinfin

120591ℎ (119905) gt 119888

(41)

The proof of uniform persistence can be done usinguniform persistence results in [31 32]

Theorem 11 If Λ gt 120574 the endemic equilibrium point E2of

system (13) is globally asymptotically stable when R119879gt 1

Proof Using the arguments in [28] system (13) satisfiesassumptions 119867(1) and 119867(2) in Γ Let 119909 = (119904

ℎ 119894ℎ 120591ℎ) and

119891(119909) be the vector field of system (13) The Jacobian matrixcorresponding to system (13) is

119869(119904ℎ 119894ℎ 120591ℎ)

=((

(

minus(120579 + Λ + 120583119867) minus120579 minus120579

Λ minus(120590

(1 + 119873119867119894ℎ)2+ 120583119867) 0

0120590

(1 + 119873119867119894ℎ)2

minus (120583119867+ 120574)

))

)

(42)

The second additive compound matrix 119869[2](119904ℎ 119894ℎ 120591ℎ)

is given by

119869[2](119904ℎ 119894ℎ120591ℎ)

=

((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) 0

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

))))

)

(43)

We let the matrix function 119875 take the form

119875 (119904ℎ 119894ℎ 120591ℎ) = diag

119894ℎ

120591ℎ

119894ℎ

120591ℎ

119894ℎ

120591ℎ

(44)

We thus have

119875119891119875minus1= diag

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(45)

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

4 Computational and Mathematical Methods in Medicine

119889119894119908

119889119905= 1205733(1 minus 119894119908) 119909 minus 120583

119882119894119908

119889119909

119889119905= 119894119908minus 120583119889119909

(14)

where = 120572119873119882119870119889 Λ = 120573

11198981(119873119882 119873119867)119894119908+ 1205732119909( + 119909)

and = 11987050119870119889 Given that the total number of bites made

by the water bugs must equal the number of bites received bythe humans119898

1(119873119882 119873119867) is a constant see [23]

3 Model Analysis

Ourmodel has two subsystems that are only coupled throughinfection term Our analysis will thus focus on the dynamicsof the environment first and then we consider how thesedynamics subsequently affect the human populationWe firstconsider the properties of the overall system before we lookat the decoupled system

31 Basic Properties Since the model monitors changes inthe populations of humans and water bugs and the densityofM ulcerans in the environment the model parameters andvariables are nonnegativeThe biologically feasible region forthe systems (13)-(14) is in R5

+and is represented by the set

Γ = (119904ℎ 119894ℎ 120591ℎ 119894119908 119909) isin R

5

+| 0 le 119904

ℎ+ 119894ℎ+ 120591ℎle 1

0 le 119894119908le 1 0 le 119909 le

120583119889

(15)

where the basic properties of local existence uniquenessand continuity of solutions are valid for the Lipschitziansystems (13)-(14)Thepopulations described in thismodel areassumed to be constant over the modelling time

We can easily establish the positive invariance of Γ Giventhat 119889119909119889119905 = 119894

119908minus 120583119889119909 le minus 120583

119889119909 we have 119909 le 120583

119889 The

solutions of systems (13)-(14) starting in Γ remain in Γ for all119905 gt 0 The 120596-limit sets of systems (13)-(14) are contained inΓ It thus suffices to consider the dynamics of our system inΓ where the model is epidemiologically and mathematicallywell posed

32 Positivity of Solutions For any nonnegative initial condi-tions of systems (13)-(14) the solutions remain nonnegativefor all 119905 isin [0infin) Here we prove that all the stated variablesremain nonnegative and the solutions of the systems (13)-(14)with nonnegative initial conditionswill remain positive for all119905 gt 0 We have the following proposition

Proposition 1 For positive initial conditions of systems (13)-(14) the solutions 119904

ℎ(119905) 119894ℎ(119905) 120591ℎ(119905) 119894119908(119905) and 119909(119905) are non-

negative for all 119905 gt 0

Proof Assume that

= sup 119905 gt 0 119904ℎgt 0 119894ℎgt 0 120591ℎgt 0 119894119908gt 0 119909 gt 0 isin (0 119905]

(16)

Thus gt 0 and it follows directly from the first equation ofthe subsystem (13) that

119889119904ℎ

119889119905le (120583119867+ 120579) minus [(120583

119867+ 120579) + Λ] 119904

ℎ (17)

This is a first order differential equation that can easily besolved using an integrating factor For a nonconstant force ofinfection Λ we have

119904ℎ() le 119904

ℎ (0) exp[minus((120583119867 + 120579) + int

0

Λ (119904) 119889119904)]

+ exp[minus((120583119867+ 120579) + int

0

Λ (119904) 119889119904)]

times [int

0

(120583119867+ 120579) 119890

((120583119867+120579)+int

0Λ(119897)119889119897)

119889]

(18)

Since the right-hand side of (18) is always positive thesolution 119904

ℎ(119905) will always be positive If Λ is constant this

result still holdsFrom the second equation of subsystem (13)

119889119894ℎ

119889119905ge minus (120583

119867+ 120590) 119894ℎge 119894ℎ (0) exp [minus (120583119867 + 120590) 119905] gt 0 (19)

The third equation of subsystem (13) yields

119889120591ℎ

119889119905ge minus (120583

119867+ 120574) 120591

ℎge 120591ℎ (0) exp [minus (120583119867 + 120574) 119905] gt 0

(20)

Similarly we can show that 119894119908(119905) gt 0 and 119909(119905) gt 0 for all 119905 gt 0

and this completes the proof

33 Environmental Dynamics The subsystem (14) representsthe dynamics of water bugs and M ulcerans in the environ-ment From the second equation we have

119909lowast=119894lowast

119908

120583119889

119894lowast

119908= 0 or 119894

lowast

119908= 1 minus

1

R119879

(21)

where

R119879=

1205733

120583119889120583119882

(22)

In this case 119909lowast = (120583119889)(1 minus (1R

119879))

The case 119894lowast119908= 0 yields the infection free equilibrium point

of the environmental dynamics submodel given by

E0= (0 0) (23)

The submodel also has an endemic equilibrium given by

E1= (120583

119882(R119879minus 1) 120583

119889120583119882(R119879minus 1)) (24)

Remark 2 It is important to note that the R119879is our model

reproduction number for the BU epidemic in the presenceof treatment driven by the dynamics of the water bug and

Computational and Mathematical Methods in Medicine 5

M ulcerans in the environment A reproduction numberusually defined as the average of the number of secondarycases generated by an index case in a naive population isa key threshold parameter that determines whether the BUdisease persists or vanishes in the population In this case itrepresents the number of secondary cases of infected waterbugs generated by the shedded M ulcerans in the environ-ment R

119879determines the infection in the environment and

subsequently in the human population We can alternativelyuse the next generation operator method [24 25] to derivethe reproduction number A similar valuewas obtained undera square root sign in this case The reproduction number isindependent of the parameters of the human population evenwhen the two submodels are combined It depends on the lifespans of the water bugs andM ulcerans in the environmentthe shedding and infection rates of the water bugs So theinfection is driven by the water bug population and thedensity of the bacterium in the environment The modelreproduction number increases linearly with the sheddingrate of theM ulcerans into the environment and the effectivecontact rate between the water bugs and M ulcerans Thisimplies that the control and management of the ulcer largelydepend on environmental management

331 Stability of E0

Theorem 3 The infection free equilibrium E0is globally stable

whenR119879lt 1 and unstable otherwise

Proof We propose a Lyapunov function of the form

V (119905) = 119894119908 +1205733

120583119889

119909 (25)

The time derivative of (25) is

V =119889119894119908

119889119905+1205733

120583119889

119889119909

119889119905

le 120583119882(R119879minus 1) 119894119908

(26)

WhenR119879le 1 V is negative and semidefinite with equality

at the infection free equilibrium andor at R119879

= 1 Sothe largest compact invariant set in Γ such that V119889119905 le 0

when R119879le 1 is the singleton E

0 Therefore by the LaSalle

Invariance Principle [26] the infection free equilibriumpointE0is globally asymptotically stable if R

119879lt 1 and unstable

otherwise

332 Stability of E1

Theorem 4 The endemic steady state E1of the subsystem (14)

is locally asymptotically stable ifR119879gt 1

Proof The Jacobian matrix of system (14) at the equilibriumpoint E

1is given by

119869E1 = (minus120583119882

1205733

minus120583119889

) (27)

Given that the trace of 119869E1 is negative and the determinant isnegative if R

119879gt 1 we can thus conclude that the unique

endemic equilibrium is locally asymptotically stable when-ever R

119879gt 1

Theorem 5 If R119879gt 1 then the unique endemic equilibrium

E1is globally stable in the interior of Γ

Proof We now prove the global stability of endemic steadystate E

1whenever it exists using the Dulac criterion and the

Poincare-Bendixson theorem The proof entails the fact thatwe begin by ruling out the existence of periodic orbits in Γusing the Dulac criteria [27] Defining the right-hand side of(14) by (119865(119894

119908 119909) 119866(119894

119908 119909)) we can construct a Dulac function

B (119894119908 119909) =

1

1205733119894119908119909 119894119908gt 0 119909 gt 0 (28)

We will thus have

120597 (119865B)

120597119894119908

+120597 (119866B)

120597119909= minus(

1

1198942119908

+

12057331199092) lt 0 (29)

Thus subsystem (14) does not have a limit cycle in Γ FromTheorem 4 ifR

119879gt 1 thenE

1is locally asymptotically stable

A simple application of the classical Poincare-Bendixsontheorem and the fact that Γ is positively invariant sufficeto show that the unique endemic steady state is globallyasymptotically stable in Γ

34 Dynamics of BU in the Human Population Our ultimateinterest is to determine how the dynamics of water bugs andM ulcerans impact the human population The overall goalis to mitigate the influence of the M ulcerans on the humanpopulation We can actually evaluate the force of infection sothat

Λ = (R119879minus 1) 120583

119882(11989811205731120583119889+

1205732

+ (R119879minus 1) 120583

119882

)

(30)

This means that the analysis of submodel (13) is subject toR119879gt 1 Our force of infection is thus now a function of

the reproduction number of submodel (14) and is constantfor any given value of the reproduction number Figure 2 is aplot of Λ versusR

119879

Using the second equation of system (13) we can evaluate119904lowast

ℎso that

119904lowast

ℎ= ([120590119894

lowast

ℎ+ 120583ℎ119894ℎ(1 + 119873

119867119894lowast

ℎ)] [ + 120583

119882(R119879minus 1)])

times ((1 + 119873119867119894lowast

ℎ) [11989811205731120583119889120583119882(R119879minus 1)

times + 120583119882(R119879minus 1)

+1205732120583119882(R119879minus 1)] )

minus1

(31)

From the third equation of (13) we have

120591lowast

ℎ=

120590119894lowastℎ

(1 + 119873119867119894lowastℎ) (120574 + 120583

119867) (32)

6 Computational and Mathematical Methods in Medicine

00 05 10 15 20000

002

004

006

008

010

RT

Λ

Figure 2 The plot of the force of infection as a function ofR119879 The

force of infection increases linearly with the reproduction numberThe human population is at risk only ifR

119879gt 1

Substituting 119904lowastℎand 120591lowastℎin the first equation of (13) at the

steady state yields a quadratic equation in 119894lowastℎgiven by

119886119894lowast2

ℎ+ 119887119894lowast

ℎ+ 119888 = 0 (33)

where

119886 = 119873119867(120583119867+ 120574) (120583

119867+ 120579)

times (1205732120583119882(R119879minus 1) + ( + (R

119879minus 1) 120583

119882)

times [120583119867+ 11989811205731120583119889120583119882(R119879minus 1)] )

119887 = 120574120579120590 + 120583119867[120579120590 + 120574 (120579 + 120590)

+120583119867(120583119867+ 120574 + 120579 + 120590)]

+ (119877119901minus 1) [ (120583

119867+ 120574) (120583

119867+ 120579) (120583

119867+ 120590)

+ 1205732(120574120579 + (120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times(120583119867+ 120579) + 120583

119867(120583119867+ 120574 + 120579 + 120590))

+ 11989811205731120583119889120574120579 +(120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times (120583119867+ 120579) + 120583

119867

times (120583119867+ 120574 + 120579 + 120590)] 120583

119882

minus 1198981(119877119901minus 1)2

1205731120583119889(minus (120579120590 + 120574 (120579 + 120590))

+ 119873119867(120574 + 120583

119867) (120579 + 120583

119867)

minus120583119867(120574 + 120579 + 120590 + 120583

119867)) 1205832

119882

119888 = minus120583119882(120583119867+ 120574) (120583

119867+ 120579)

times [1205732+ 11989811205731120583119889( + 120583

119882(R119879minus 1))] (R

119879minus 1)

(34)

Clearly our model has two possible steady states given by

E119886

2= (119904lowast

ℎ 119894lowast+

ℎ 120591lowast

ℎ) E

119887

2= (119904lowast

ℎ 119894lowastminus

ℎ 120591lowast

ℎ) (35)

where 119894lowastplusmnℎ

are roots of the quadratic equation (33) We notethat if R

119879gt 1 we have 119886 gt 0 and 119888 lt 0 By Descartesrsquo rule

of signs irrespective of the sign of 119887 the quadratic equation(33) has one positive root the endemic equilibrium E119886

2= E2

We thus have the following result

Theorem 6 System (13) has a unique endemic equilibrium E2

wheneverR119879gt 1

Remark 7 It is important to note that when subsystem (14) isat its infection free steady state then the human populationwill also be free of the BUWe can easily establish the BU freeequilibrium in humans as Eℎ

0= (1 0 0) The existence of Eℎ

0

is thus subject to the water bugs and the environment beingfree ofM ulcerans

341 Local Stability of Eℎ0

Theorem 8 The disease free equilibrium Eℎ0whenever it exists

is locally asymptotically stable if R119879

lt 1 and unstableotherwise

Proof When R119879lt 1 then either there are no infections in

the water bugs or they are simply carriers So Eℎ0exists The

Jacobian matrix of system (13) at the disease free equilibriumpoint Eℎ

0is given by

119869Eℎ0

= (

minus (120583119867+ 120579) minus120579 minus120579

0 minus (120583119867+ 120590) 0

0 120590 minus (120583119867+ 120574)

) (36)

The eigenvalues of 119869Eℎ0

are 1205821= minus(120583

119867+120579) 120582

2= minus(120583

119867+120590) and

1205823= minus(120583

119867+ 120574) We can thus conclude that the disease free

equilibrium is locally asymptotically stable whenever R119879lt

1

342 Local Stability of E2

Theorem 9 The unique endemic equilibrium point E2is

locally asymptotically stable for R119879gt 1

Proof The Jacobian matrix at the endemic steady state E2is

given by

119869E2 =(

(

minus(120583119867+ 120579) minus Λ minus120579 minus120579

Λ minus120583119867minus

120590

(1 + 119894lowastℎ)2

0

0120590

(1 + 119894lowastℎ)2

minus (120583119867+ 120574)

)

)

(37)

Computational and Mathematical Methods in Medicine 7

If we let 120595 = 120590(1 + 119894lowastℎ)2 then the eigenvalues of 119869E2 are given

by the solutions of the characteristic polynomial

1205993+ 12057811205992+ 1205782120599 + 1205783= 0 (38)

where

1205781= (120583119867+ 120574) + (120583

119867+ 120579) + (120583

119867+ 120595) + Λ

1205782= (120583119867+ 120579) (120583

119867+ 120574Λ) + (120583

119867+ 120574) (120583

119867+ 120595 + Λ)

+ (120583119867+ 120579) (120583

119867+ 120595) + Λ120595

1205783= 120579Λ (120574 + 120595) + 120574 (120579 + Λ) 120595

+ 120583119867(Λ120595 + 120579 (Λ + 120595) + 120574 (120579 + Λ + 120595)

+120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

(39)

Using the Routh-Hurwitz criterion we note that 1205781gt

0 1205782gt 0 and 120578

3gt 0 The evaluation of 120578

11205782minus 1205783yields

(120579 + Λ) (120579 + 120595) (Λ + 120595) + 1205742(120579 + Λ + 120595) + 120574(120579 + Λ + 120595)

2

+ 2120583119867(1205742+ 1205792+ Λ2+ 3Λ120595 + 120595

2+ 3120579 (Λ + 120595)

+3120574 (120579 + Λ + 120595) + 4120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

gt 0

(40)

This establishes the necessary and sufficient conditions for allroots of the characteristic polynomial to lie on the left half ofthe complex plane So the endemic equilibrium E

2is locally

asymptotically stable

In the next section we establish the global stability of theendemic equilibrium using the approach according to Li andMuldowney [28] based on monotone dynamical systems andoutlined in Appendix A of [29 30]

343 Global Stability of the Endemic Equilibrium We beginby stating the following theorem

Theorem 10 If R119879gt 1 system (13) is uniformly persistent in

Γ the interior of ΓThe existence of Eℎ

0 only if R

119879gt 1 guarantees uniform

persistence [31] System (13) is said to be uniformly persistentif there exists a positive constant 119888 such that any solution(119904ℎ(119905) 119894ℎ(119905) 120591ℎ(119905)) with initial conditions (119904

ℎ(0) 119894ℎ(0) 120591ℎ(0)) isin

Γ satisfies

lim inf119905rarrinfin

119904ℎ (119905) gt 119888 lim inf

119905rarrinfin

119894ℎ (119905) gt 119888

lim inf119905rarrinfin

120591ℎ (119905) gt 119888

(41)

The proof of uniform persistence can be done usinguniform persistence results in [31 32]

Theorem 11 If Λ gt 120574 the endemic equilibrium point E2of

system (13) is globally asymptotically stable when R119879gt 1

Proof Using the arguments in [28] system (13) satisfiesassumptions 119867(1) and 119867(2) in Γ Let 119909 = (119904

ℎ 119894ℎ 120591ℎ) and

119891(119909) be the vector field of system (13) The Jacobian matrixcorresponding to system (13) is

119869(119904ℎ 119894ℎ 120591ℎ)

=((

(

minus(120579 + Λ + 120583119867) minus120579 minus120579

Λ minus(120590

(1 + 119873119867119894ℎ)2+ 120583119867) 0

0120590

(1 + 119873119867119894ℎ)2

minus (120583119867+ 120574)

))

)

(42)

The second additive compound matrix 119869[2](119904ℎ 119894ℎ 120591ℎ)

is given by

119869[2](119904ℎ 119894ℎ120591ℎ)

=

((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) 0

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

))))

)

(43)

We let the matrix function 119875 take the form

119875 (119904ℎ 119894ℎ 120591ℎ) = diag

119894ℎ

120591ℎ

119894ℎ

120591ℎ

119894ℎ

120591ℎ

(44)

We thus have

119875119891119875minus1= diag

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(45)

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

Computational and Mathematical Methods in Medicine 5

M ulcerans in the environment A reproduction numberusually defined as the average of the number of secondarycases generated by an index case in a naive population isa key threshold parameter that determines whether the BUdisease persists or vanishes in the population In this case itrepresents the number of secondary cases of infected waterbugs generated by the shedded M ulcerans in the environ-ment R

119879determines the infection in the environment and

subsequently in the human population We can alternativelyuse the next generation operator method [24 25] to derivethe reproduction number A similar valuewas obtained undera square root sign in this case The reproduction number isindependent of the parameters of the human population evenwhen the two submodels are combined It depends on the lifespans of the water bugs andM ulcerans in the environmentthe shedding and infection rates of the water bugs So theinfection is driven by the water bug population and thedensity of the bacterium in the environment The modelreproduction number increases linearly with the sheddingrate of theM ulcerans into the environment and the effectivecontact rate between the water bugs and M ulcerans Thisimplies that the control and management of the ulcer largelydepend on environmental management

331 Stability of E0

Theorem 3 The infection free equilibrium E0is globally stable

whenR119879lt 1 and unstable otherwise

Proof We propose a Lyapunov function of the form

V (119905) = 119894119908 +1205733

120583119889

119909 (25)

The time derivative of (25) is

V =119889119894119908

119889119905+1205733

120583119889

119889119909

119889119905

le 120583119882(R119879minus 1) 119894119908

(26)

WhenR119879le 1 V is negative and semidefinite with equality

at the infection free equilibrium andor at R119879

= 1 Sothe largest compact invariant set in Γ such that V119889119905 le 0

when R119879le 1 is the singleton E

0 Therefore by the LaSalle

Invariance Principle [26] the infection free equilibriumpointE0is globally asymptotically stable if R

119879lt 1 and unstable

otherwise

332 Stability of E1

Theorem 4 The endemic steady state E1of the subsystem (14)

is locally asymptotically stable ifR119879gt 1

Proof The Jacobian matrix of system (14) at the equilibriumpoint E

1is given by

119869E1 = (minus120583119882

1205733

minus120583119889

) (27)

Given that the trace of 119869E1 is negative and the determinant isnegative if R

119879gt 1 we can thus conclude that the unique

endemic equilibrium is locally asymptotically stable when-ever R

119879gt 1

Theorem 5 If R119879gt 1 then the unique endemic equilibrium

E1is globally stable in the interior of Γ

Proof We now prove the global stability of endemic steadystate E

1whenever it exists using the Dulac criterion and the

Poincare-Bendixson theorem The proof entails the fact thatwe begin by ruling out the existence of periodic orbits in Γusing the Dulac criteria [27] Defining the right-hand side of(14) by (119865(119894

119908 119909) 119866(119894

119908 119909)) we can construct a Dulac function

B (119894119908 119909) =

1

1205733119894119908119909 119894119908gt 0 119909 gt 0 (28)

We will thus have

120597 (119865B)

120597119894119908

+120597 (119866B)

120597119909= minus(

1

1198942119908

+

12057331199092) lt 0 (29)

Thus subsystem (14) does not have a limit cycle in Γ FromTheorem 4 ifR

119879gt 1 thenE

1is locally asymptotically stable

A simple application of the classical Poincare-Bendixsontheorem and the fact that Γ is positively invariant sufficeto show that the unique endemic steady state is globallyasymptotically stable in Γ

34 Dynamics of BU in the Human Population Our ultimateinterest is to determine how the dynamics of water bugs andM ulcerans impact the human population The overall goalis to mitigate the influence of the M ulcerans on the humanpopulation We can actually evaluate the force of infection sothat

Λ = (R119879minus 1) 120583

119882(11989811205731120583119889+

1205732

+ (R119879minus 1) 120583

119882

)

(30)

This means that the analysis of submodel (13) is subject toR119879gt 1 Our force of infection is thus now a function of

the reproduction number of submodel (14) and is constantfor any given value of the reproduction number Figure 2 is aplot of Λ versusR

119879

Using the second equation of system (13) we can evaluate119904lowast

ℎso that

119904lowast

ℎ= ([120590119894

lowast

ℎ+ 120583ℎ119894ℎ(1 + 119873

119867119894lowast

ℎ)] [ + 120583

119882(R119879minus 1)])

times ((1 + 119873119867119894lowast

ℎ) [11989811205731120583119889120583119882(R119879minus 1)

times + 120583119882(R119879minus 1)

+1205732120583119882(R119879minus 1)] )

minus1

(31)

From the third equation of (13) we have

120591lowast

ℎ=

120590119894lowastℎ

(1 + 119873119867119894lowastℎ) (120574 + 120583

119867) (32)

6 Computational and Mathematical Methods in Medicine

00 05 10 15 20000

002

004

006

008

010

RT

Λ

Figure 2 The plot of the force of infection as a function ofR119879 The

force of infection increases linearly with the reproduction numberThe human population is at risk only ifR

119879gt 1

Substituting 119904lowastℎand 120591lowastℎin the first equation of (13) at the

steady state yields a quadratic equation in 119894lowastℎgiven by

119886119894lowast2

ℎ+ 119887119894lowast

ℎ+ 119888 = 0 (33)

where

119886 = 119873119867(120583119867+ 120574) (120583

119867+ 120579)

times (1205732120583119882(R119879minus 1) + ( + (R

119879minus 1) 120583

119882)

times [120583119867+ 11989811205731120583119889120583119882(R119879minus 1)] )

119887 = 120574120579120590 + 120583119867[120579120590 + 120574 (120579 + 120590)

+120583119867(120583119867+ 120574 + 120579 + 120590)]

+ (119877119901minus 1) [ (120583

119867+ 120574) (120583

119867+ 120579) (120583

119867+ 120590)

+ 1205732(120574120579 + (120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times(120583119867+ 120579) + 120583

119867(120583119867+ 120574 + 120579 + 120590))

+ 11989811205731120583119889120574120579 +(120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times (120583119867+ 120579) + 120583

119867

times (120583119867+ 120574 + 120579 + 120590)] 120583

119882

minus 1198981(119877119901minus 1)2

1205731120583119889(minus (120579120590 + 120574 (120579 + 120590))

+ 119873119867(120574 + 120583

119867) (120579 + 120583

119867)

minus120583119867(120574 + 120579 + 120590 + 120583

119867)) 1205832

119882

119888 = minus120583119882(120583119867+ 120574) (120583

119867+ 120579)

times [1205732+ 11989811205731120583119889( + 120583

119882(R119879minus 1))] (R

119879minus 1)

(34)

Clearly our model has two possible steady states given by

E119886

2= (119904lowast

ℎ 119894lowast+

ℎ 120591lowast

ℎ) E

119887

2= (119904lowast

ℎ 119894lowastminus

ℎ 120591lowast

ℎ) (35)

where 119894lowastplusmnℎ

are roots of the quadratic equation (33) We notethat if R

119879gt 1 we have 119886 gt 0 and 119888 lt 0 By Descartesrsquo rule

of signs irrespective of the sign of 119887 the quadratic equation(33) has one positive root the endemic equilibrium E119886

2= E2

We thus have the following result

Theorem 6 System (13) has a unique endemic equilibrium E2

wheneverR119879gt 1

Remark 7 It is important to note that when subsystem (14) isat its infection free steady state then the human populationwill also be free of the BUWe can easily establish the BU freeequilibrium in humans as Eℎ

0= (1 0 0) The existence of Eℎ

0

is thus subject to the water bugs and the environment beingfree ofM ulcerans

341 Local Stability of Eℎ0

Theorem 8 The disease free equilibrium Eℎ0whenever it exists

is locally asymptotically stable if R119879

lt 1 and unstableotherwise

Proof When R119879lt 1 then either there are no infections in

the water bugs or they are simply carriers So Eℎ0exists The

Jacobian matrix of system (13) at the disease free equilibriumpoint Eℎ

0is given by

119869Eℎ0

= (

minus (120583119867+ 120579) minus120579 minus120579

0 minus (120583119867+ 120590) 0

0 120590 minus (120583119867+ 120574)

) (36)

The eigenvalues of 119869Eℎ0

are 1205821= minus(120583

119867+120579) 120582

2= minus(120583

119867+120590) and

1205823= minus(120583

119867+ 120574) We can thus conclude that the disease free

equilibrium is locally asymptotically stable whenever R119879lt

1

342 Local Stability of E2

Theorem 9 The unique endemic equilibrium point E2is

locally asymptotically stable for R119879gt 1

Proof The Jacobian matrix at the endemic steady state E2is

given by

119869E2 =(

(

minus(120583119867+ 120579) minus Λ minus120579 minus120579

Λ minus120583119867minus

120590

(1 + 119894lowastℎ)2

0

0120590

(1 + 119894lowastℎ)2

minus (120583119867+ 120574)

)

)

(37)

Computational and Mathematical Methods in Medicine 7

If we let 120595 = 120590(1 + 119894lowastℎ)2 then the eigenvalues of 119869E2 are given

by the solutions of the characteristic polynomial

1205993+ 12057811205992+ 1205782120599 + 1205783= 0 (38)

where

1205781= (120583119867+ 120574) + (120583

119867+ 120579) + (120583

119867+ 120595) + Λ

1205782= (120583119867+ 120579) (120583

119867+ 120574Λ) + (120583

119867+ 120574) (120583

119867+ 120595 + Λ)

+ (120583119867+ 120579) (120583

119867+ 120595) + Λ120595

1205783= 120579Λ (120574 + 120595) + 120574 (120579 + Λ) 120595

+ 120583119867(Λ120595 + 120579 (Λ + 120595) + 120574 (120579 + Λ + 120595)

+120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

(39)

Using the Routh-Hurwitz criterion we note that 1205781gt

0 1205782gt 0 and 120578

3gt 0 The evaluation of 120578

11205782minus 1205783yields

(120579 + Λ) (120579 + 120595) (Λ + 120595) + 1205742(120579 + Λ + 120595) + 120574(120579 + Λ + 120595)

2

+ 2120583119867(1205742+ 1205792+ Λ2+ 3Λ120595 + 120595

2+ 3120579 (Λ + 120595)

+3120574 (120579 + Λ + 120595) + 4120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

gt 0

(40)

This establishes the necessary and sufficient conditions for allroots of the characteristic polynomial to lie on the left half ofthe complex plane So the endemic equilibrium E

2is locally

asymptotically stable

In the next section we establish the global stability of theendemic equilibrium using the approach according to Li andMuldowney [28] based on monotone dynamical systems andoutlined in Appendix A of [29 30]

343 Global Stability of the Endemic Equilibrium We beginby stating the following theorem

Theorem 10 If R119879gt 1 system (13) is uniformly persistent in

Γ the interior of ΓThe existence of Eℎ

0 only if R

119879gt 1 guarantees uniform

persistence [31] System (13) is said to be uniformly persistentif there exists a positive constant 119888 such that any solution(119904ℎ(119905) 119894ℎ(119905) 120591ℎ(119905)) with initial conditions (119904

ℎ(0) 119894ℎ(0) 120591ℎ(0)) isin

Γ satisfies

lim inf119905rarrinfin

119904ℎ (119905) gt 119888 lim inf

119905rarrinfin

119894ℎ (119905) gt 119888

lim inf119905rarrinfin

120591ℎ (119905) gt 119888

(41)

The proof of uniform persistence can be done usinguniform persistence results in [31 32]

Theorem 11 If Λ gt 120574 the endemic equilibrium point E2of

system (13) is globally asymptotically stable when R119879gt 1

Proof Using the arguments in [28] system (13) satisfiesassumptions 119867(1) and 119867(2) in Γ Let 119909 = (119904

ℎ 119894ℎ 120591ℎ) and

119891(119909) be the vector field of system (13) The Jacobian matrixcorresponding to system (13) is

119869(119904ℎ 119894ℎ 120591ℎ)

=((

(

minus(120579 + Λ + 120583119867) minus120579 minus120579

Λ minus(120590

(1 + 119873119867119894ℎ)2+ 120583119867) 0

0120590

(1 + 119873119867119894ℎ)2

minus (120583119867+ 120574)

))

)

(42)

The second additive compound matrix 119869[2](119904ℎ 119894ℎ 120591ℎ)

is given by

119869[2](119904ℎ 119894ℎ120591ℎ)

=

((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) 0

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

))))

)

(43)

We let the matrix function 119875 take the form

119875 (119904ℎ 119894ℎ 120591ℎ) = diag

119894ℎ

120591ℎ

119894ℎ

120591ℎ

119894ℎ

120591ℎ

(44)

We thus have

119875119891119875minus1= diag

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(45)

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

6 Computational and Mathematical Methods in Medicine

00 05 10 15 20000

002

004

006

008

010

RT

Λ

Figure 2 The plot of the force of infection as a function ofR119879 The

force of infection increases linearly with the reproduction numberThe human population is at risk only ifR

119879gt 1

Substituting 119904lowastℎand 120591lowastℎin the first equation of (13) at the

steady state yields a quadratic equation in 119894lowastℎgiven by

119886119894lowast2

ℎ+ 119887119894lowast

ℎ+ 119888 = 0 (33)

where

119886 = 119873119867(120583119867+ 120574) (120583

119867+ 120579)

times (1205732120583119882(R119879minus 1) + ( + (R

119879minus 1) 120583

119882)

times [120583119867+ 11989811205731120583119889120583119882(R119879minus 1)] )

119887 = 120574120579120590 + 120583119867[120579120590 + 120574 (120579 + 120590)

+120583119867(120583119867+ 120574 + 120579 + 120590)]

+ (119877119901minus 1) [ (120583

119867+ 120574) (120583

119867+ 120579) (120583

119867+ 120590)

+ 1205732(120574120579 + (120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times(120583119867+ 120579) + 120583

119867(120583119867+ 120574 + 120579 + 120590))

+ 11989811205731120583119889120574120579 +(120574 + 120579) 120590 minus 119873

119867(120583119867+ 120574)

times (120583119867+ 120579) + 120583

119867

times (120583119867+ 120574 + 120579 + 120590)] 120583

119882

minus 1198981(119877119901minus 1)2

1205731120583119889(minus (120579120590 + 120574 (120579 + 120590))

+ 119873119867(120574 + 120583

119867) (120579 + 120583

119867)

minus120583119867(120574 + 120579 + 120590 + 120583

119867)) 1205832

119882

119888 = minus120583119882(120583119867+ 120574) (120583

119867+ 120579)

times [1205732+ 11989811205731120583119889( + 120583

119882(R119879minus 1))] (R

119879minus 1)

(34)

Clearly our model has two possible steady states given by

E119886

2= (119904lowast

ℎ 119894lowast+

ℎ 120591lowast

ℎ) E

119887

2= (119904lowast

ℎ 119894lowastminus

ℎ 120591lowast

ℎ) (35)

where 119894lowastplusmnℎ

are roots of the quadratic equation (33) We notethat if R

119879gt 1 we have 119886 gt 0 and 119888 lt 0 By Descartesrsquo rule

of signs irrespective of the sign of 119887 the quadratic equation(33) has one positive root the endemic equilibrium E119886

2= E2

We thus have the following result

Theorem 6 System (13) has a unique endemic equilibrium E2

wheneverR119879gt 1

Remark 7 It is important to note that when subsystem (14) isat its infection free steady state then the human populationwill also be free of the BUWe can easily establish the BU freeequilibrium in humans as Eℎ

0= (1 0 0) The existence of Eℎ

0

is thus subject to the water bugs and the environment beingfree ofM ulcerans

341 Local Stability of Eℎ0

Theorem 8 The disease free equilibrium Eℎ0whenever it exists

is locally asymptotically stable if R119879

lt 1 and unstableotherwise

Proof When R119879lt 1 then either there are no infections in

the water bugs or they are simply carriers So Eℎ0exists The

Jacobian matrix of system (13) at the disease free equilibriumpoint Eℎ

0is given by

119869Eℎ0

= (

minus (120583119867+ 120579) minus120579 minus120579

0 minus (120583119867+ 120590) 0

0 120590 minus (120583119867+ 120574)

) (36)

The eigenvalues of 119869Eℎ0

are 1205821= minus(120583

119867+120579) 120582

2= minus(120583

119867+120590) and

1205823= minus(120583

119867+ 120574) We can thus conclude that the disease free

equilibrium is locally asymptotically stable whenever R119879lt

1

342 Local Stability of E2

Theorem 9 The unique endemic equilibrium point E2is

locally asymptotically stable for R119879gt 1

Proof The Jacobian matrix at the endemic steady state E2is

given by

119869E2 =(

(

minus(120583119867+ 120579) minus Λ minus120579 minus120579

Λ minus120583119867minus

120590

(1 + 119894lowastℎ)2

0

0120590

(1 + 119894lowastℎ)2

minus (120583119867+ 120574)

)

)

(37)

Computational and Mathematical Methods in Medicine 7

If we let 120595 = 120590(1 + 119894lowastℎ)2 then the eigenvalues of 119869E2 are given

by the solutions of the characteristic polynomial

1205993+ 12057811205992+ 1205782120599 + 1205783= 0 (38)

where

1205781= (120583119867+ 120574) + (120583

119867+ 120579) + (120583

119867+ 120595) + Λ

1205782= (120583119867+ 120579) (120583

119867+ 120574Λ) + (120583

119867+ 120574) (120583

119867+ 120595 + Λ)

+ (120583119867+ 120579) (120583

119867+ 120595) + Λ120595

1205783= 120579Λ (120574 + 120595) + 120574 (120579 + Λ) 120595

+ 120583119867(Λ120595 + 120579 (Λ + 120595) + 120574 (120579 + Λ + 120595)

+120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

(39)

Using the Routh-Hurwitz criterion we note that 1205781gt

0 1205782gt 0 and 120578

3gt 0 The evaluation of 120578

11205782minus 1205783yields

(120579 + Λ) (120579 + 120595) (Λ + 120595) + 1205742(120579 + Λ + 120595) + 120574(120579 + Λ + 120595)

2

+ 2120583119867(1205742+ 1205792+ Λ2+ 3Λ120595 + 120595

2+ 3120579 (Λ + 120595)

+3120574 (120579 + Λ + 120595) + 4120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

gt 0

(40)

This establishes the necessary and sufficient conditions for allroots of the characteristic polynomial to lie on the left half ofthe complex plane So the endemic equilibrium E

2is locally

asymptotically stable

In the next section we establish the global stability of theendemic equilibrium using the approach according to Li andMuldowney [28] based on monotone dynamical systems andoutlined in Appendix A of [29 30]

343 Global Stability of the Endemic Equilibrium We beginby stating the following theorem

Theorem 10 If R119879gt 1 system (13) is uniformly persistent in

Γ the interior of ΓThe existence of Eℎ

0 only if R

119879gt 1 guarantees uniform

persistence [31] System (13) is said to be uniformly persistentif there exists a positive constant 119888 such that any solution(119904ℎ(119905) 119894ℎ(119905) 120591ℎ(119905)) with initial conditions (119904

ℎ(0) 119894ℎ(0) 120591ℎ(0)) isin

Γ satisfies

lim inf119905rarrinfin

119904ℎ (119905) gt 119888 lim inf

119905rarrinfin

119894ℎ (119905) gt 119888

lim inf119905rarrinfin

120591ℎ (119905) gt 119888

(41)

The proof of uniform persistence can be done usinguniform persistence results in [31 32]

Theorem 11 If Λ gt 120574 the endemic equilibrium point E2of

system (13) is globally asymptotically stable when R119879gt 1

Proof Using the arguments in [28] system (13) satisfiesassumptions 119867(1) and 119867(2) in Γ Let 119909 = (119904

ℎ 119894ℎ 120591ℎ) and

119891(119909) be the vector field of system (13) The Jacobian matrixcorresponding to system (13) is

119869(119904ℎ 119894ℎ 120591ℎ)

=((

(

minus(120579 + Λ + 120583119867) minus120579 minus120579

Λ minus(120590

(1 + 119873119867119894ℎ)2+ 120583119867) 0

0120590

(1 + 119873119867119894ℎ)2

minus (120583119867+ 120574)

))

)

(42)

The second additive compound matrix 119869[2](119904ℎ 119894ℎ 120591ℎ)

is given by

119869[2](119904ℎ 119894ℎ120591ℎ)

=

((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) 0

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

))))

)

(43)

We let the matrix function 119875 take the form

119875 (119904ℎ 119894ℎ 120591ℎ) = diag

119894ℎ

120591ℎ

119894ℎ

120591ℎ

119894ℎ

120591ℎ

(44)

We thus have

119875119891119875minus1= diag

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(45)

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

Computational and Mathematical Methods in Medicine 7

If we let 120595 = 120590(1 + 119894lowastℎ)2 then the eigenvalues of 119869E2 are given

by the solutions of the characteristic polynomial

1205993+ 12057811205992+ 1205782120599 + 1205783= 0 (38)

where

1205781= (120583119867+ 120574) + (120583

119867+ 120579) + (120583

119867+ 120595) + Λ

1205782= (120583119867+ 120579) (120583

119867+ 120574Λ) + (120583

119867+ 120574) (120583

119867+ 120595 + Λ)

+ (120583119867+ 120579) (120583

119867+ 120595) + Λ120595

1205783= 120579Λ (120574 + 120595) + 120574 (120579 + Λ) 120595

+ 120583119867(Λ120595 + 120579 (Λ + 120595) + 120574 (120579 + Λ + 120595)

+120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

(39)

Using the Routh-Hurwitz criterion we note that 1205781gt

0 1205782gt 0 and 120578

3gt 0 The evaluation of 120578

11205782minus 1205783yields

(120579 + Λ) (120579 + 120595) (Λ + 120595) + 1205742(120579 + Λ + 120595) + 120574(120579 + Λ + 120595)

2

+ 2120583119867(1205742+ 1205792+ Λ2+ 3Λ120595 + 120595

2+ 3120579 (Λ + 120595)

+3120574 (120579 + Λ + 120595) + 4120583119867(120574 + 120579 + Λ + 120595 + 120583

119867))

gt 0

(40)

This establishes the necessary and sufficient conditions for allroots of the characteristic polynomial to lie on the left half ofthe complex plane So the endemic equilibrium E

2is locally

asymptotically stable

In the next section we establish the global stability of theendemic equilibrium using the approach according to Li andMuldowney [28] based on monotone dynamical systems andoutlined in Appendix A of [29 30]

343 Global Stability of the Endemic Equilibrium We beginby stating the following theorem

Theorem 10 If R119879gt 1 system (13) is uniformly persistent in

Γ the interior of ΓThe existence of Eℎ

0 only if R

119879gt 1 guarantees uniform

persistence [31] System (13) is said to be uniformly persistentif there exists a positive constant 119888 such that any solution(119904ℎ(119905) 119894ℎ(119905) 120591ℎ(119905)) with initial conditions (119904

ℎ(0) 119894ℎ(0) 120591ℎ(0)) isin

Γ satisfies

lim inf119905rarrinfin

119904ℎ (119905) gt 119888 lim inf

119905rarrinfin

119894ℎ (119905) gt 119888

lim inf119905rarrinfin

120591ℎ (119905) gt 119888

(41)

The proof of uniform persistence can be done usinguniform persistence results in [31 32]

Theorem 11 If Λ gt 120574 the endemic equilibrium point E2of

system (13) is globally asymptotically stable when R119879gt 1

Proof Using the arguments in [28] system (13) satisfiesassumptions 119867(1) and 119867(2) in Γ Let 119909 = (119904

ℎ 119894ℎ 120591ℎ) and

119891(119909) be the vector field of system (13) The Jacobian matrixcorresponding to system (13) is

119869(119904ℎ 119894ℎ 120591ℎ)

=((

(

minus(120579 + Λ + 120583119867) minus120579 minus120579

Λ minus(120590

(1 + 119873119867119894ℎ)2+ 120583119867) 0

0120590

(1 + 119873119867119894ℎ)2

minus (120583119867+ 120574)

))

)

(42)

The second additive compound matrix 119869[2](119904ℎ 119894ℎ 120591ℎ)

is given by

119869[2](119904ℎ 119894ℎ120591ℎ)

=

((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) 0

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

))))

)

(43)

We let the matrix function 119875 take the form

119875 (119904ℎ 119894ℎ 120591ℎ) = diag

119894ℎ

120591ℎ

119894ℎ

120591ℎ

119894ℎ

120591ℎ

(44)

We thus have

119875119891119875minus1= diag

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(45)

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

8 Computational and Mathematical Methods in Medicine

where 119875119891is the diagonal element matrix derivative of 119875 with

respect to time and

119875119869[2]119875minus1=

(((((

(

minus[120579 + Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] 0 120579

120590

(1 + 119873119867119894ℎ)2

minus (120579 + Λ + 2120583119867+ 120574) minus120579

0 Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2)

)))))

)

(46)

where 1015840 represents the derivative with respect to timeThematrix119876 = 119875

119891119875minus1+119875119869[2]119875minus1 can be written as a block

matrix so that

119876 = (11987611

11987612

11987621

11987622

) (47)

where

11987611= minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

11987612= (0 120579) 119876

21= (

120590

(1 + 119873119867119894ℎ)2

0

)

11987622=(

minus(120579 + Λ + 2120583119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

minus120579

Λ minus(2120583119867+ 120574 +

120590

(1 + 119873119867119894ℎ)2) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

)

(48)

Let (119909 119910 119911) denote the vectors in R3 and let the norm in R3

be defined by1003816100381610038161003816(119909 119910 119911)

1003816100381610038161003816 = max |119909| 1003816100381610038161003816119910 + 1199111003816100381610038161003816 (49)

Also let L denote the Lozinski i measure with respect tothis norm Following [33] we have

L (119876) le sup 1198921 1198922

equiv sup L1(11987611) +

1003816100381610038161003816119876121003816100381610038161003816 L1 (11987622) +

1003816100381610038161003816119876211003816100381610038161003816

(50)

where |11987612| and |119876

21| are thematrix normswith respect to the

vector norm 1198711 andL1is the Lozinski i measure with respect

to the 1198711 normIn fact

L1(11987611) = minus[120579 + Λ + 2120583

119867+

120590

(1 + 119873119867119894ℎ)2] +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

1003816100381610038161003816119876121003816100381610038161003816 = 120579

1003816100381610038161003816119876211003816100381610038161003816 =

120590

(1 + 119873119867119894ℎ)2

L1(11987622) = minus (120579 + 2120583

119867+ 120574) +

1198941015840ℎ

119894ℎ

minus1205911015840ℎ

120591ℎ

(51)

We now have

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

(52)

The third equation of (13) gives

1205911015840ℎ

120591ℎ

= (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574) (53)

Substituting (53) into (52) yields

1198921=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2] minus

1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus [Λ + 2120583119867+

120590

(1 + 119873119867119894ℎ)2]

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

Computational and Mathematical Methods in Medicine 9

minus (120590

(1 + 119873119867119894ℎ))(

119894ℎ

120591ℎ

) minus (120583119867+ 120574)

le1198941015840ℎ

119894ℎ

minus Λ minus 120574 + 120583119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1198922=1198941015840ℎ

119894ℎ

minus (120579 + 2120583119867+ 120574) +

120590

(1 + 119873119867119894ℎ)2minus1205911015840ℎ

120591ℎ

=1198941015840ℎ

119894ℎ

minus (120579 + 120583119867) + (

120590

(1 + 119873119867119894ℎ))(

1

(1 + 119873119867119894ℎ)minus119894ℎ

120591ℎ

)

le1198941015840ℎ

119894ℎ

minus [(120579 + 120583119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)]

(54)

where 119888 is the constant of uniform persistence The inequali-ties followTheorem 10

If we impose the condition Λ gt 120574 then

L (119876) le sup 1198921 1198922

=1198941015840ℎ

119894ℎ

minus 120596(55)

where 120596 = min1205961 1205962 with

1205961= Λ minus 120574 + 120583

119867+

120590

(1 + 119873119867119888)2+ (

120590

(1 + 119873119867119888)2)

1205962= (120579 + 120583

119867) minus (

120590

(1 + 119873119867119888))(

1

(1 + 119873119867119888)minus 1)

(56)

Hence

1

119905int119905

0

L (119876) 119889119904 le1

119905log

119894ℎ (119905)

119894ℎ (0)

minus 120596 (57)

The imposed condition implies that the infection rate isgreater than the recovery rateThe result follows based on theBendixson criterion proved in [28]

4 Numerical Simulations

In this section we endeavour to give some simulation resultsfor the combined subsystems (13) and (14) The simulationsare performed using MALAB and we set our time in yearsWe carry out sensitivity analysis to determine the effectsof a chosen parameter on the state variables Specificallywe chose to focus on the parameters that make up themodel reproduction number because we are interested inparameters that aid the reduction of the BU epidemic Wenow give a brief exposition on parameter estimation

41 Parameter Estimation The estimation of parameters inthe model validation process is a challenging process Wemake some hypothetical assumptions for the purpose of

illustrating the usefulness of our model in tracking thedynamics of the BU Demographic parameters are the easiestto estimate For the mortality rate 120583

119867 we assume that the

life expectancy of the human population is 61 years Thisvalue has been the approximation of the life expectancy inGhana [34] and is indeed applicable to sub-Saharan AfricaThis translates into 120583

119867= 00166 per year or equivalently

45 times 10minus5 per day The Buruli ulcer is a vector borne disease

and some of the parameters we have can be estimated fromliterature on vector borne diseases Recovery rates of vectorborne diseases range from 16 times 10

minus5 to 05 per day [35] Therate of loss of immunity 120579 for vector borne diseases rangesbetween 0 and 11times10minus2 per day [35] Although themortalityrate of the water bugs is not known it is assumed to be015 per day [18] We assume that we have more water bugsthan humans so that 119898

1gt 1 The remaining parameters

were reasonably estimated based on literature on vector bornediseases and the intuitive understanding of the BU disease bythe first two authors

42 Sensitivity Analysis Many of the parameters used in thispaper are not experimentally obtained It is thus important totest how these parameters affect the output of the variablesThis is achieved by employing sensitivity and uncertaintyanalysis techniques In this subsection we explore the sen-sitivity analysis of the model parameters to find out thedegree to which the parameters influence the outputs ofthe model We determine the partial correlation coefficients(PRCCs) of the parameters The parameters with negativePRCCs reduce the severity of the BU epidemic while thosewith positive PRCCs aggravate it Using Latin hypercubesampling (LHS) scheme with 1000 simulations for each runwe investigate only four of the most significant parametersThese parameters influence only submodel (14) The scatterplots are shown in Figure 3

Figures 3(a) and 3(b) depict parameters with a posi-tive correlation with the reproduction number They showa monotonic increase of R

119879as 120572 and 120573

3increase This

means that to curtail the epidemic the reduction in theshedding rate and infection of water bugs by M ulcerans isof paramount importance On the other hand Figures 3(c)and 3(d) show a negative correlation with the reproductionnumber This means that the clearance of the water bug andtheM ulcerans in the environment will reduce the spread ofBU epidemic

A more informative comparison of how the parametersinfluence the model is given in Figure 4 The tornado plotshows that the parameter 120572 affects the reproduction morethan any of the other parameters considered So interventionstargeted towards the reduction in the shedding rate of Mulcerans into the environment will significantly slow theepidemic

43 Simulation Results To validate our mathematical anal-ysis results we plot phase diagrams for R

119879less than 1 and

greater than 1 for the environmental dynamics The globalproperties of the steady states are confirmed in Figures 5(a)and 5(b)The black dots show the location of the steady states

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

10 Computational and Mathematical Methods in Medicine

07 08 085065 075

Shedding rate of Mulcerans 120572

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(a)

05505 06506 07507 08508 09509

Effective contact rate for water bugs 1205733

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(b)

04504 05 06 07 08055 065 075

Clearance rate of Mulcerans 120583d

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(c)

04504 05 06 07055 065 075

Clearance rate of water bugs 120583W

minus06

minus04

minus02

0

02

04

06

08

1

12

14

log(RT

)

(d)

Figure 3 The scatter plots for the parameters 120572 1205733 120583119889 and 120583

119882

020 04 06 08 1

120572

minus04 minus02

1205733

120583d

120583W

Figure 4 The tornado plots for the four parameters in the model reproduction number

Figure 6 shows a three-dimensional phase diagram forthe human population dynamics The existence of theendemic equilibrium when R

119879gt 1 is numerically shown

here The plot shows the trajectories of parametric solutionsof (13) for randomly chosen initial conditionsThe position ofthe endemic equilibrium point is indicated on the diagram

To determine how the infection of the water bugs trans-lates into the transmission of BU in humans we plot thefraction of BU in humans over time while varying 120583

119889 the

clearance rate of bacteria from the environment Figure 7(a)shows how the infections of BU in humans change withvariations in the value of 120583

119889 The infections are evaluated as

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

Computational and Mathematical Methods in Medicine 11

0 1 2

1

2

x(t)

iw(t)

(a)

0 1 2

1

2

3

4

x(t)

iw(t)

(b)

Figure 5 The phase diagrams for R119879= 08889 (a) and R

119879= 53333 (b)

005

1

00020040060080

002

004

006

008

01

012

014

016

Susceptible

Infective

In tr

eatm

ent

Endemic steady state

01

Figure 6 A phase diagram for the human population showing the endemic steady state For a randomly chosen set of initial conditions alltrajectories tend to an endemic equilibrium for the following parameter values 120583

119867= 002 120579 = 004 Λ = 007 120590 = 04 and 120574 = 07

the number of infected humansThe figure shows that as theclearance of the bacteria from the environment increases thistranslates into a reduction in the number of infected humancases Similar results can be obtained if the parameter 120583

119908

is considered as shown in Figure 7(b) So interventions toreduce the impact of the epidemic on humans can also beinstituted through the reduction of bacteria andwater bugs inthe environment It is important to note that the practicalityof such an intervention is a mirageThis has worked for othervector borne diseases such as malaria

We also explore the role played by direct M ulceransinfection on the transmission dynamics of BU Researchhas shown that antecedent trauma has often been related tothe lesions that characterize BU [36] Figure 8 shows howthe proportion of infected humans changes with increasingtransmission rate through direct contact with the environ-ment The figures show that as the transmission rate ofdirect contact of humans with the environment increases

the proportion of the infected also increases This has directimplications on how humans interact with the environment

The shedding rate of M ulcerans into the environmentand the treatment rate of humans are important to considerIn Figure 9(a) we observe that increasing the amount of Mulcerans shed in the environment increases the infections ofthe BU disease in the human population While this maysound very obvious the quantification of the effects thereofis of particular significance We also note that there arebenefits in increasing the maximum threshold with regard totreatment An increase in the value of 120590will lead to a decreasein the number of infections in the human population

5 Conclusion

We present a deterministic model whose main aim wasto capture the two potential routes of transmission andtreatment uptake in a resource limited population This

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

12 Computational and Mathematical Methods in Medicine

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

120583d = 002

120583d = 00198120583d = 00196

120583d = 00194

(a)

0 55 11 165 22 275 33 385 44 495 55Time (years)

001

0012

0014

0016

0018

002

0022

Infe

cted

hum

ans

120583W = 0025

120583W = 00248

120583W = 00246

120583W = 00244

(b)

Figure 7 Fraction of the infected human population for R119879= 16492 for the parameters 120583

119889and 120583

119882 The parameter values used for the

constant parameters are 120583119867= 000045 119898

1= 10 120579 = 0011 120574 = 0000016 120573

3= 009 120590 = 008 = 04000 119873

119867= 100000 120572 =

000615 1205731= 000001 and 120573

2= 00000002

0 55 11 165 22 275 33 385 44 495 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

1205732 = 2 times 10minus7

1205732 = 2 times 10minus61205732 = 1 times 10minus5

1205732 = 2 times 10minus5

Figure 8 The proportion of infected humans for the given values of 1205732and the following parameter values 120583

119867= 000045 119898

1= 10 120579 =

0011 120574 = 0000016 1205733= 009 120583

119889= 002 120590 = 014 119870 = 04000 120583

119882= 0025 119873

119867= 100000 120572 = 0006 and 120573

1= 000001

uptake is not linear and hence we propose a responsefunction of theMichaelis-Menten type Because of the natureof the infection process our model is divided into twosubmodels that are only coupled through the infection termThe model is analysed by determining the steady states Theanalysis is done through the submodels The model in thispaper presented a unique challenge in which the infectionin one submodel takes place at the steady state of the othersubmodel The model analysis is carried out in terms of

the model reproduction number R119879 Numerical simulations

are carried out The model parameters were estimated fromliterature and sensitivity analysis was done because not muchof the disease is understood and parameter estimation wasdifficult Through the simulations changes in the numberof BU cases in the human population were determined fordifferent values of the clearance rates of the water bugsand M ulcerans Our main result is that the managementof BU depends mostly on the environmental management

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

Computational and Mathematical Methods in Medicine 13

0 55 11 22 33 44 55001

0012

0014

0016

0018

002

0022

Time (years)

Infe

cted

hum

ans

165 275 385 495

120572 = 0006

120572 = 000605

120572 = 00061

120572 = 000615

(a)

001

0011

0012

0013

0014

0015

0016

0017

0018

0019

002

Infe

cted

hum

ans

0 55 11 22 33 44 55Time (years)

165 275 385 495

120590 = 008 120590 = 012

120590 = 01 120590 = 014

(b)

Figure 9 A phase diagram for the infected water bugs and M ulcerans in the environment for the same parameters presented in Figure 7with 120583

119889= 002 and 120583

119882= 0025

that is clearance of the bacteria from the environment andreduction in shedding This in turn will reduce the infectionof the water bugs that transmit the infection to humans Asmentioned earlier clearance of M ulcerans and water bugsis not practically feasible but non the less very informativeResearch in malaria now looks at vector control sterilisationand genetic modification of the mosquito Such an approachcould be beneficial with regard to the control of water bugs

This model presents the very few attempts to mathe-matically model BU A lot of additional extensions can bemadeThemodel can be transformed into a delay differentialequation dynamical system to capture treatment delays thatare often fatal to BU victims Social interventions such aseducational campaigns can be included in the model tocapture various campaigns and initiatives to stop the diseaseFinally this model can be used to suggest the type of data thatshould be collected as research on the ulcer intensifies

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E Bonyah and I Dontwi acknowledge with thanks the sup-port of the Department of Mathematics Kwame NkrumahUniversity of Science and Technology F Nyabadza acknowl-edges with gratitude the support from theNational ResearchFoundation South Africa and Post Graduate and Interna-tional Office Stellenbosch University for the production ofthis paper

References

[1] J van Ravensway M E Benbow A A Tsonis et al ldquoClimateand landscape factors associated with Buruli ulcer incidence invictoria Australiardquo PLoS ONE vol 7 no 12 Article ID e510742012

[2] World Health Organization ldquoBuruli ulcer disease (Mycobac-terium ulcerans infection) Fact Sheet Nu199rdquoWorld Health Or-ganization 2007 httpwwwwhointmediacentrefact-sheetsfs199en

[3] RWMerritt E DWalker P L C Small J RWallace and P DR Johnson ldquoEcology and transmission of Buruli ulcer diseaseA systematic reviewrdquo PLoS Neglected Tropical Diseases vol 4no 12 article e911 2010

[4] A A Duker F Portaels and M Hale ldquoPathways of Mycobac-terium ulcerans infection a reviewrdquo Environment Internationalvol 32 no 4 pp 567ndash573 2006

[5] M Wansbrough-Jones and R Phillips ldquoBuruli ulcer emergingfrom obscurityrdquo The Lancet vol 367 no 9525 pp 1849ndash18582006

[6] D S Walsh F Portaels and W M Meyers ldquoBuruli ulcer(Mycobacterium ulcerans infection)rdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 102 no 10 pp969ndash978 2008

[7] KAsiedu and S Etuaful ldquoSocioeconomic implications of Buruliulcer in Ghana a three-year reviewrdquo The American Journal ofTropical Medicine and Hygiene vol 59 no 6 pp 1015ndash10221998

[8] A A Duker E J M Carranza and M Hale ldquoSpatial relation-ship between arsenic in drinking water and Mycobacteriumulcerans infection in the Amansie West district Ghanardquo Min-eralogical Magazine vol 69 no 5 pp 707ndash717 2005

[9] T Wagner M E Benbow T O Brenden J Qi and RC Johnson ldquoBuruli ulcer disease prevalence in Benin WestAfrica associations with land usecover and the identification

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003

14 Computational and Mathematical Methods in Medicine

of disease clustersrdquo International Journal of Health Geographicsvol 7 article 25 2008

[10] S A Al-Sheikh ldquoModelling and analysis of an SEIR epidemicmodel with a limited resource for treatmentrdquo Global Journal ofScience Frontier Research Mathematics and Decision Sciences vol 12 no 14 pp 56ndash66 2012

[11] P Agbenorku I K Donwi P Kuadzi and P SaundersonldquoBuruli Ulcer treatment challenges at three Centres in GhanardquoJournal of Tropical Medicine vol 2012 Article ID 371915 7pages 2012

[12] C K Ahorlu E Koka D Yeboah-Manu I Lamptey and EAmpadu ldquoEnhancing Buruli ulcer control in Ghana throughsocial interventions a case study from the Obom sub-districtrdquoBMC Public Health vol 13 no 1 article 59 2013

[13] P J Converse E L Nuermberger D V Almeida and J HGrosset ldquoTreating Mycobacterium ulcerans disease (Buruliulcer) from surgery to antibiotics is the pill mightier than thekniferdquo Future Microbiology vol 6 no 10 pp 1185ndash1198 2011

[14] X Zhang and X Liu ldquoBackward bifurcation of an epidemicmodel with saturated treatment functionrdquo Journal ofMathemat-ical Analysis and Applications vol 348 no 1 pp 433ndash443 2008

[15] J Gao and M Zhao ldquoStability and bifurcation of an epidemicmodel with saturated treatment functionrdquo Communications inComputer and Information Science vol 234 no 4 pp 306ndash3152011

[16] H R Williamson M E Benbow L P Campbell et al ldquoDetec-tion of Mycobacterium ulcerans in the environment predictsprevalence of Buruli ulcer in Beninrdquo PLoS Neglected TropicalDiseases vol 6 no 1 Article ID e1506 2012

[17] G E Sopoh Y T Barogui R C Johnson et al ldquoFamilyrelationship water contact and occurrence of buruli ulcer inBeninrdquo PLoS Neglected Tropical Diseases vol 4 no 7 articlee746 2010

[18] A Y Aidoo and B Osei ldquoPrevalence of aquatic insects andarsenic concentration determine the geographical distributionofMycobacterium ulceran infectionrdquo Computational and Math-ematical Methods in Medicine vol 8 no 4 pp 235ndash244 2007

[19] M T Silva F Portaels and J Pedrosa ldquoAquatic insects andMycobacterium ulcerans an association relevant to buruli ulcercontrolrdquo PLoS Medicine vol 4 no 2 article e63 2007

[20] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[21] F Nyabadza ldquoOn the complexities of modeling HIVAIDS inSouthern AfricardquoMathematical Modelling and Analysis vol 13no 4 pp 539ndash552 2008

[22] N K Martin P Vickerman and M Hickman ldquoMathematicalmodelling of hepatitis C treatment for injecting drug usersrdquoJournal of Theoretical Biology vol 274 pp 58ndash66 2011

[23] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[24] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[25] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[26] J P LaSalle The Stability of Dynamical Systems CBMS-NSFRegional Conference Series in Applied Mathematics 25 SIAMPhiladelphia Pa USA 1976

[27] J K Hale Ordinary Differential Equations John Wiley amp SonsNew York NY USA 1969

[28] M Y Li and J S Muldowney ldquoA geometric approach to global-stability problemsrdquo SIAM Journal onMathematical Analysis vol27 no 4 pp 1070ndash1083 1996

[29] B Buonomo A drsquoOnofrio and D Lacitignola ldquoGlobal stabilityof an SIR epidemicmodel with information dependent vaccina-tionrdquoMathematical Biosciences vol 216 no 1 pp 9ndash16 2008

[30] H Yang H Wei and X Li ldquoGlobal stability of an epidemicmodel for vector-borne diseaserdquo Journal of Systems Science ampComplexity vol 23 no 2 pp 279ndash292 2010

[31] H I Freedman S G Ruan and M X Tang ldquoUniformpersistence and flows near a closed positively invariant setrdquoJournal of Dynamics and Differential Equations vol 6 no 4 pp583ndash601 1994

[32] M Y Li J R Graef LWang and J Karsai ldquoGlobal dynamics ofa SEIR model with varying total population sizerdquoMathematicalBiosciences vol 160 no 2 pp 191ndash213 1999

[33] J Martin ldquoLogarithmic norms and projections applied to lineardifferential systemsrdquo Journal of Mathematical Analysis andApplications vol 45 pp 432ndash454 1974

[34] Ghana Statistical Service 2014 httpwwwstatsghanagovgh[35] G Rascalou D Pontier F Menu and S Gourbiere ldquoEmergence

and prevalence of human vector-borne diseases in sink vectorpopulationsrdquo PLoS ONE vol 7 no 5 Article ID e36858 2012

[36] M Debacker C Zinsou J Aguiar W M Meyers and FPortaels ldquoFirst case of Mycobacterium ulcerans disease (Buruliulcer) following a human biterdquo Clinical Infectious Diseases vol36 no 5 pp e67ndashe68 2003