On the Transmission Dynamics of Buruli Ulcer in Ghana: Insights Through a Mathematical Model Farai Nyabadza1 and Ebenezer Bonyah2*

Nyabadza and Bonyah BMC Res Notes (2015) 8:656 DOI 10.1186/s13104-015-1619-5

RESEARCH ARTICLE Open Access On the transmission dynamics of Buruli ulcer in Ghana: Insights through a mathematical model Farai Nyabadza1 and Ebenezer Bonyah2*

Abstract Background: Mycobacterium ulcerans is know to cause the Buruli ulcer. The association between the ulcer and environmental exposure has been documented. However, the epidemiology of the ulcer is not well understood. A hypothesised transmission involves humans being bitten by the water bugs that prey on mollusks, snails and young fishes. Methods: In this paper, a model for the transmission of Mycobacterium ulcerans to humans in the presence of a preventive strategy is proposed and analysed. The model equilibria are determined and conditions for the existence of the equilibria established. The model analysis is carried out in terms of the reproduction number R0. The disease free equilibrium is found to be locally asymptotically stable for R0 < 1. The model is fitted to data from Ghana. Results: The model is found to exhibit a backward bifurcation and the endemic equilibrium point is globally stable when R0 > 1. Sensitivity analysis showed that the Buruli ulcer epidemic is highly influenced by the shedding and clearance rates of Mycobacterium ulcerans in the environment. The model is found to fit reasonably well to data from Ghana and projections on the future of the Buruli ulcer epidemic are also made. Conclusions: The model reasonably fitted data from Ghana. The fitting process showed data that appeared to have reached a steady state and projections showed that the epidemic levels will remain the same for the projected time. The implications of the results to policy and future management of the disease are discussed. Keywords: Buruli ulcer, Transmission dynamics, Basic reproduction number, Sensitivity analysis, Stability

Background This disease has dramatically emerged in several west Buruli ulcer is caused by pathogenic bacterium where African countries, such as Ghana, Cote d’Ivoire, Benin, infection often leads to extensive destruction of skin and and Togo in recent years [26]. soft tissue through the formation of large ulcers usu- The transmission mode of the ulcer is not well under- ally on the legs or arms [28]. It is a devastating disease stood, however residence near an aquatic environment caused by Mycobacterium ulcerans. The ulcer is fast has been identified as a risk factor for the ulcer in Africa becoming a debilitating affliction in many countries [3]. [6, 16, 25]. Transmission is thus likely to occur through It is named after a region called Buruli, near the Nile contact with the environment [20]. Recent studies in West River in Uganda, where in 1961 the first large number of Africa have implicated aquatic bugs as transmission vec- cases was reported. In Africa, close to 30,000 cases were tors for the ulcer [18, 24]. An attractive hypothesis for a reported between 2005 and 2010 [29]. Cote d’Ivoire, with possible mode of transmission to humans was proposed the highest incidence, reported 2533 cases in 2010 [27]. by Portaels et al. [22]: water-filtering hosts (fish, mollusks) concentrate the Mycobacterium ulcerans bacteria present *Correspondence: [email protected] in water or mud and discharge them again to this environ- 2 Department of Mathematics and Statistics, Kumasi Polytechnic, P. O. ment, where they are then ingested by aquatic predators Box 854, Kumasi, Ghana Full list of author information is available at the end of the article such as beetles and water bugs. These insects, in turn,

© 2015 Nyabadza and Bonyah. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons. org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 2 of 15

may transmit the disease to humans by biting [18]. Per- model. We also determine the steady states and ana- son to person transmission is less likely. Aquatic bugs are lysed their stability. The results of this paper are given in insects found throughout temperate and tropical envi- “Results”. Parameter estimation, sensitivity analysis and ronments with abundant freshwater. They prey, accord- the numerical results on the behavior of the model are ing to their size, on mollusks, snails, young fishes, and also presented in this section. The paper is concluded in the adults and larvae of other insects that they capture “Discussion”. with their raptorial front legs and bite with their rostrum. These insects can inflict painful bites on humans as well. Methods In Ghana, where Buruli ulcer is endemic, the water bugs Model formulation are present in swamps and rivers, where human activities We consider a constant human population NH (t), the such as farming, fishing, and bathing take place [18]. vector population of water bugs NV (t) and the fish pop- Research on Buruli ulcer has focused mainly on the ulation NF (t) at any time t. The total human population socio-cultural aspects of the disease. The research recom- is divided into three epidemiological subclasses of those mends the need for Information, Education and Commu- that are susceptible SH (t), the infected IH (t) and the nication (IEC) intervention strategies, to encourage early recovered who are still immune RH (t). Total population case detection and treatment with the assumption that of vector (water bug) at any time t is divided into two sub- once people gain knowledge they will take the appropri- classes to susceptible water bugs SV (t) and those that are ate action to access treatment early [2]. IEC is defined as infectious and can transmit the Buruli ulcer to humans, an approach which attempts to change or reinforce a set IV (t). The total population reservoir of small fish is also of behaviours to a targeted group regarding a problem. divided into two compartments of susceptible fishS F (t) The IEC strategy is preventive in that it has a potential of and infected fish IF (t). We also consider the role of the enhancing control of the ulcer [5]. It is also important to environment by introducing a compartment U, repre- note that Buruli ulcer is treatable with antibiotics. A com- senting the density of Mycobacterium ulcerans in the bination of rifampin and streptomycin administered daily environment. We make the following basic assumptions: for 8 weeks has the potential to eliminate Mycobacterium ulcerans bacilli and promote healing without relapse. •• Mycobacterium ulcerans are transferred only from Mathematical models have been used to model the vector ( water bug) to the humans. transmission of many diseases globally. Many advances in •• There is homogeneity of human, water bug and fish the management of diseases have been born from math- populations’ interactions. ematical modeling [11, 12, 14, 15]. Mathematical models •• Infected humans recover and are temporarily can evaluate actual or potential control measures in the immune, but lose immunity. absence of experiments, see for instance [19]. To the best •• Fish are preyed on by the water bugs. of our knowledge very few mathematical models have •• Unlike some bacterial infections such as leprosy been formulated to analyse the transmission dynamics (caused by Mycobacterium leprae) and tuberculosis of Mycobacterium ulcerans. This could be largely due to (caused by Mycobacterium tuberculosis), which are the elusive epidemiology of the Buruli ulcer. Aidoo and characterized by person-to-person contact transmis- Osei [3] proposed a mathematical model of the SIR-type sion, it is hypothesized that Mycobacterium ulcerans in an endeavour to explain the transmission of Myco- is acquired through environmental contact and direct bacterium ulcerans and its dependence on arsenic. In person-to-person transmission is rare [20]. this paper, we propose a model which takes into account •• Susceptible host (human population) can be infected the human population, water bugs as vectors and fish as through biting by an infectious vector (water bug). potential reservoirs of Mycobacterium ulcerans follow- We represent the effective biting rate that an infec- ing the transmission dynamics described in [8]. In addi- tious vector has to susceptible host as βH and tion we include the preventive control measures in a bid the incidence of new infections transmitted by to capture the IEC strategy. Our main aim is to study the water bugs is expressed by standard incidence rate dynamics of the Buruli ulcer in the presence of a preven- SH IV βH . One can interpret βH as a function of tive control strategy, while emphasizing the role of the NH vector (water bugs) and fish and their interaction with the biting frequency of the infected water bugs on the environment. The model is then validated using data humans, density of infectious water bugs per human, from Ghana. This is crucial in informing policy and sug- the probability that a bite will result in an infection gesting strategies for the control of the disease. and the efficacy of the IEC strategy. In particular we This paper is arranged as follows; in “Methods”, we βH (1 ǫ)τ αβ , ǫ (0, 1) can set = − H∗ where ∈ is the formulate and establish the basic properties of the efficacy of the IEC strategy,τ the number of water Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 3 of 15

bugs per human host, α the biting frequency (the bit- New susceptibles enter the population at a rate of ing rate of humans by a single water bug) and βH∗ the µH NH . Buruli ulcer sufferers do not recover with per- probability that a bite by an infected vector to a sus- manent immunity, they loose immunity at a rate θ and

ceptible human will produce an infection. SV IF become susceptible again. Susceptibles and infected •• Susceptible water bugs are infected at a rate βV through interaction with infected water bugs, with infec- NV tion driven by water bugs biting susceptible humans. SV U ηvβV Once infected, individuals are allowed to recover either through predation of infected fish and K spontaneously or through antibiotic treatment at a rate representing other sources in the environment. Here γ . In this model, the human population is assumed to be constant over the modeling time with the birth and ηV differentiates the infectivity potential of the fish SV from that of the environment. death rates being equal. The compartment tracks the changes in the susceptible water bugs population that are •• Assuming fish prey on infected water bugs, sus- µ N SF IV recruited at a rate V V. The infection of water bugs is ceptible fish are infected at a rateβ F through NF driven by two processes: their interaction infected fish and SF U with the environment. The natural mortality of the water ηF βF µ . S predation of infected fish and K represent- bugs occurs at a rate V Similarly, the compartment F ing infection through the environment. Here ηF is tracks the changes in the susceptible fish population that µ N a modification parameter that models the relative are recruited at a rate F F. The infection of fish is also infectivity of fish from that of the environment. driven by two processes: their interaction infected water •• The vector population and the fish populations are bugs and with the environment. Fish’s natural mortality µ . assumed to be constant. The growth functions are rate is F The growth of Mycobacterium ulcerans in the respectively given by g(NV ) and g(NF ), where environment is driven by their shedding by infected water bugs and fish into the environment. They are assumed g(NV ) µV NV and g(NF ) µF NF . µ . = = to die naturally at a rate E The possible interrelations It is important to note that other types of functions between humans, the water bug and fish are represented can be chosen as growth functions. In this work we by the schematic diagram below (Fig. 1). however assume that the growth functions are linear. The descriptions of the parameters that describe the •• There is a proposed hypothesis that environmen- flow rates between compartments are given in Table 1. tal mycobacteria in the bottoms of swamps may be The dynamics of the ulcer can be described by the fol- mechanically concentrated by small water-filtering lowing set of nonlinear differential equations: organisms such as microphagous fish, snails, mos- dSH SH IV µH NH θRH βH µH SH , quito larvae, small crustaceans, and protozoa [8]. We dt N  = + − H −  assume that fish increase the environmental con- dI S I  σ . H H V I  centrations of Mycobacterium ulcerans at a rate F βH (µH γ) H ,  dt = NH − +  Humans are are assumed not to shed any bacteria  dRH  into the environment. γ IH (µH θ)RH ,  dt  •• Aquatic bugs release bacteria into the environment at = − +  dS S I S U  a rate σV . V V F V  µV NV βV ηV βV µV SV ,  dt = − NV − K −  •• The model does not include a potential route of  (1) direct contact with the bacterium in water. dIV SV IF SV U  βV ηV βV µV IV , •• The birth rate of the human population is directly dt = NV + K −   proportional to the size of the human population. dSF SF IV SF U  µF NF βF ηF βF µF SF ,  •• The recovery of infected individuals is assumed to dt N K  = − F − −  occur both spontaneously and through treatment. dI S I S U  F F V F I  Research has shown that localized lesions may spon- βF ηF βF µF F ,  dt = NF + K −  taneously heal but, without treatment, most cases of  dU  Buruli ulcer result in physical deformities that often σF IF σV IV µEU.  dt = + −  lead to physiological abnormalities and stigmas [4]. 

We now describe briefly, the transmission dynamics of We assume that all the model parameters are positive and Buruli ulcer: the initial conditions of the model system (1) are given by Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 4 of 15

Fig. 1 Proposed transmission dynamics of the Buruli ulcer among humans, fish, water bugs and the environment (U)

Table 1 Description of parameters used in the model SH (0) SH0 > 0, IH (0) IH0 0, RH (0) RH0 0, = = ≥ = = SV (0) SV 0 > 0, Symbol Description = IV (0) IV 0 0, SF (0) SF0 > 0, βH The effective contact rate between the vector and susceptible = ≥ = IF (0) IF0 0 and U(0) U0 > 0. humans = ≥ = βV The effective contact rate between fish and susceptible vectors 1 βF The effective contact rate between the susceptible fish and Myco- We arbitrarily scale the time t by the quantity µV by let- τ µV t bacterium ulcerans ting = and introduce the following dimensionless γ The recovery rate of infected humans parameters; θ The rate of loss of immunity of recovered humans µH Natural mortality rate/birth rate of the human population βH µH θ γ µV Natural mortality rate of the vector population τ = µV t, βh = , µh = , θh = , γh = , µV µV µV µV µF Natural mortality rate of the fish population NH NF rV The growth rate of the vector population m = , m = , 1 N 2 N rF The growth rate of the fish population V V N N K The environmental carrying capacity of the bacteria population 1 F V µF βF m3 = , m4 = , m5 = , µf = , βf = , σF Rate of shedding of Mycobacterium ulcerans into the environment m2 K K µV µV by fish σF σV βV µE σV Rate of shedding of Mycobacterium ulcerans into the environment σf = , σv = , βv = and µe = . by the water bugs µV µV µV µV µE Rate at which Mycobacterium ulcerans are cleared from the environment Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 5 of 15

can be non dimensionalised bySo, system (1) setting system (2). The populations described in this model are SH IH RH IV assumed to be constant over the modelling time. The sh = , ih = , rh = , iv = , NH NH NH NV solutions of system (2) starting in remain in for all t > 0. Thus , is positively invariant and it is sufficient to SF IF U sf = , if = and u = . consider solutions in . NF NF K The forces of infection for humans, water bugs and fish Positivity of solutions are respectively We desire to show that for any non-negative initial conditions of system (2), say (SH0, IH0, IV 0, IF0, U0), the solu- H = βhm1iv, V = βvm2if + ηV βvu, τ 0, ). tions remain non-negative for all ∈[ ∞ We prove F = βf m3iv + ηF βf u. that all the state variables remain non-negative and the solutions of the system (2) with positive initial conditions Given that the total number of bites made by the water will remain positive for all τ>0. We thus state the fol- bugs must equal the number of bites received by the lowing lemma. humans, m1 is a constant, see [9]. Similarly m2 is constant and so is m3. We also note that since NF and NV are con- Lemma 1 Given that the initial conditions of system (2) stants, m4 and m5 are constants. are positive, the solutions SH (τ), IH (τ), IV (τ), IF (τ) and s i r 1, sv iv 1, s i 1 U(τ) τ>0 Given that h + h + h = + = f + f = are non-negative for all . 0 u 1, and ≤ ≤ system (1) can be reduced to the following system of equations by conveniently maintaining the cap- Proof Assume that italised subscripts so that we can still respectively write τ sup τ>0 SH > 0, IH > 0, IV > 0, IF > 0, U > 0 (0, τ . sh, ih, iv, if and u as SH , IH , IV , IF and U. ˆ = { : } ∈ ] τ>0, dSH Thus ˆ and it follows directly from the first equation (µ θ )(1 SH ) θ IH H SH , dτ = h + h − − h −  of the system (2) that   dS dI  H H  (θh H )SH . H SH (µh γh)IH ,  dτ ≥− + dτ = − +    We thus have dIV  V (1 IV ) µvIV ,  (2) τ dτ = − −  dSH SH0 exp θht H (ς)dς. dt ≥ − +  0 dIF  F (1 IF ) µf IF ,  Since the exponential function is always positive and dτ = − −   SH0 SH (0)>0, SH (τ)  = the solution will thus be always dU  positive. m4σ IF m5σvIV µeU.  dτ = f + −   From the second equation of (2), Basic properties dIH (µh γh)IH , Feasible region dτ ≥− + dU (µ γ )τ m4σf IF m5σvIV µeU m4σf IH IH0e− h+ h > 0. Note that dτ = + − ≤ ⇒ ≥ m U IV (τ) > 0, IF (τ) > 0 5σv µe . Through integration we obtain Similarly, it can be shown that and + − U(τ) > 0 τ>0, m4σf m5σv for all and this completes the proof. U + . ≤ µ The feasible region (the region where e Steady states analysis the model makes biological sense) for the system (2) is in The disease free equilibrium R5 and is represented by the set + In this section, we solve for the equilibrium points by setting the right hand side of system (2) to zero. This direct S I I I U R5 S I � ( H , H , V , F , ) 0 H H 1, calculation shows that system (2) always has a disease = ∈ +| ≤ + ≤ m4σf m5σv free equilibrium point 0 IV 1, 0 IF 1, 0 U + , ≤ ≤ ≤ ≤ ≤ ≤ µe E0 (1, 0, 0, 0, 0). = We have the following result on the local stability of the where the basic properties of local existence, uniqueness disease free equilibrium. and continuity of solutions are valid for the Lipschitzian Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 6 of 15

E Theorem 1 The disease free equilibrium 0 whenever The endemic equilibrium R it exists, is locally asymptotically stable if 0 < 1 and The endemic equilibrium is much more tedious to obtain. ∗ β m1I∗ , unstable otherwise. Given that H = h V from the first and second equations of system (2) we have Proof The Jacobian matrix of system (2) at the equilib- 1 m1βhIV∗ E0 S∗ and I∗ , rium point is given by H = 1 AI H = (µ γ )(1 AI ) + V∗ h + h + V∗ (µh θh) θh m1βh 00  − + − −  m1βh(µh θh γh) 0 (µh γh) m1βh 00 A + + . J − + m where = (µ γ )(µ θ ) E0  001 2βv ηvβv . h h h h =  −  + +  00m3βf µf ηf βf   −  The last equation of system (2) can be written as  00m5σv m4σf µe  − m4σf m5σv U ∗ ϑ1I∗ ϑ2I∗ , where ϑ1 and ϑ2 . = F + V = µ = µ It can be seen that the eigenvalues of JE0 are e e (µ θ ), (µ γ ) − h + h − h + h and the solution of the charac- We thus have teristic polynomial and 3 2 F∗ ϑ3IV∗ ϑ4IF V∗ ϑ5IV∗ ϑ6IF∗, P(ϑ) ϑ a2ϑ a1ϑ µeµ (1 R0) 0, = + = + = + + + f − = where ϑ3 = βf (m3 + ϑ2ηf ), ϑ4 = ϑ1βf ηf , ϑ5 = ϑ2ηvβv and where ϑ6 = βv(m2 + ϑ1ηv).

a2 1 µe µf , From the third and fourth equations of system (2)we = + + have a1 µe µf µeµf (βf βv m4ηf σf βf m5ηvσvβv) and = + + − + + I 1 ϑ (1 I ) R R1 R2 R3, V∗ 5 V∗ 0 0 0 0 IF∗ [ − − ], (3) = + + = ϑ6(1 I ) − V∗ for IF∗ µf ϑ4(1 IF∗) I∗ [ − − ]. m4ηf σf βf m5ηvσvβv V (4) 1 = 2 = = ϑ3(1 IF∗) R0 , R0 and − µeµf µe Substituting (3) into (4) we obtain IV∗ 0 and the cubic µe + m m ηvσf + m m ηf σv = 3 3 4 2 5 equation R0 = βf βv . µeµf 3 2 f (I∗ ) a3I∗ a2I∗ a1I∗ a0 0, (5) V = V + V + V + = The solutions of P(ϑ) 0 have negative real parts only if R = where 0 < 1 following the use of the Routh Hurwitz Criterion.

We can thus conclude that the disease free equilibrium is βf µf R a m m R locally asymptotically stable whenever 0 < 1. 0 µe 2 4ηvσf [ 0 1], R = µe + − We note that 0 is the model system (2)’s reproduction a1 ϑ4ϑ5(1 ϑ6) ϑ5(ϑ4 ϑ3ϑ6) ϑ3ϑ5ϑ6 number and does not depend on the human population = + + + + 1 2 , size. The model reproduction number is a sum of three ϑ3ϑ6( ϑ6) ϑ5(ϑ4ϑ5 µf ϑ6) ϑ4ϑ5 1 2 −[ + + + + ] R R a2 (1 ϑ6)(ϑ4 ϑ3ϑ6) ϑ5(ϑ4ϑ5 µf ϑ6) terms. The terms 0 and 0 represent the contribution of = + + + + fish and water bugs respectively to the infection dynam- ϑ6(ϑ3ϑ6 µf ϑ5) 2ϑ4ϑ5(1 ϑ6) ϑ6(ϑ3ϑ5 µf ) , 3 + + −[ + + + ] R , 2 ics. The term 0 which is not very common in many epi- m5βf ηvσvβ a v m m m m m 0. demiological models, shows the combined contribution 3 2 (µe 2 4ηvσf ) 3 2 5ηf σv < =− µe + + of the water bugs, fish and their shedding of Mycobac- terium ulcerans into the environment. So, the infection Note that is driven by the water bugs, fish and the density of the > 0 if R > 1 bacterium in the environment. The model reproduc- a 0 0 < 0 if R < 1. tion number increases linearly with the shedding rates 0 of the Mycobacterium ulcerans into the environment by Given that βf fish and water bugs and the effective contact rates and 2 β f ′(I∗ ) 3a3(I∗ ) 2a2 ∗ a1, (6) v . It decreases with increasing removal rates of the fish V = V + 1 + and Mycobacterium ulcerans. So the control of the ulcer the turning points of equation (5) are given by depends largely on environmental management. Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 7 of 15

2 Theorem 2 The model system (2) has a2 a 3a1a3 1,2 2 (7) (I∗ ) − ± − . R > 1 V = 3a 1 a unique endemic equilibrium point if 0 . 3 Rc R 2 has two endemic equilibria for 0 < 0 < 1 where Rc 2 0 is the critical threshold below which no endemic The discriminant of solutions (7) is a2 3a1a3. We △= − equilibrium exists. now focus on the sign of the discriminant. < 0 f (I ) Rc If △ , then V∗ has no real turning points, which Remark The evaluation of 0 depends on the signs of implies that f (IV∗ ) is a strictly monotonic function. The a2 and a1 and the sign of the discriminant. The compu- sign of f ′( 1∗) is crucial in determining the monotonicity. tations are algebraically involving and long and are not Through completing the square, equation (6) can be writ- included here. Since the model system (2) possesses two Rc R ten as endemic equilibria when 0 < 0 < 1, the model exhib- R its backward bifurcation for 0 < 1. a 2 1 2 2 f ′(IV∗ ) 3a3 IV∗ 2 (3a1a3 a2) . The consequence of the above remark is that bringing = + 3a3 + 9a − R 3 0 below unity is not sufficient to eradicate the disease. R (8) For eradication, 0 must be brought below the critical Rc 2 value 0. < 0 3a1a3 a > 0 a3 < 0 Clearly if △ , then − 2 . Since , then f (I )<0 f (I ) ′ V∗ . Thus V∗ is a strictly monotone decreas- Global stability of the endemic equilibrium ( ) ing function. Note that limI∗ f IV∗ . For V →∓∞ = ±∞ f (0) a0 < 0, the polynomial f (IV∗ ) has no positive real E = R Theorem 3 The endemic equilibrium point 1 of system roots for 0 < 1,. However, if f (0) a0 > 0 it has only R = (2), is globally asymptotically stable. one positive real root for 0 > 1, and consequently only one endemic equilibrium. Proof The global stability of the endemic equilibrium, If 0, then f ′(IV∗ ) has only one real root with mul- △= 1 2 a2 (I∗ ) (I∗ ) can be determined by constructing a Lyapunov function tiplicity two. This implies that V V 3a3 and V = =− (t) such that that f ′(IV∗ )<0. Thus the polynomial f (IV∗ ) is a decreas- a2 f ′′(I∗ )( ) 0, ing function. Given that V − 3a3 = the turning f (I ). SH IH point is a point of inflexion for V∗ The polynomial V(t) S S S ln A I I I ln H H∗ H∗ S H H∗ H∗ I f (IV∗ ) has only one endemic equilibrium. = − − H∗ + − − H∗ For > 0, we consider two cases; a1 < 0 and a1 > 0 . IV IF △ B IV I I ln C IF I I ln a1 < 0 a1a3 > 0 √ < a2 V∗ V∗ I F∗ F∗ I If , then . This means that △ . Irre- + − − V∗ + − − F∗ spective of the sign of a2, f ′(IV∗ ) has two real positive U D U U U ln . and distinct roots. This implies that (5) has two posi- ∗ ∗ U (9) + − − ∗ f (0) a0 > 0 R0 > 1 tive turning points. If = i.e then, f (I∗ ) V has at least one positive real root, and hence at V(t) least one endemic equilibrium. On the other hand, if The corresponding time derivative of is given by f (0) a0 < 0 then, f (IV∗ ) has at most two positive real = R SH∗ IH∗ IV∗ roots when 0 < 1, and hence at most two endemic V 1 SH A 1 IH B 1 IV ˙ = − S ˙ + − I ˙ + − I ˙ equilibria. H H V > < √ > I∗ U ∗ If a1 0, then a1a3 0, which implies that a2 . C 1 F I D 1 U. △ ˙F ˙ (10) For a2 > 0, f ′(IV∗ ) has two real roots of opposite signs. + − IF + − U R Since f (0) a0 > 0 for 0 > 1, then, f (IV∗ ) has one pos- = At the endemic equilibrium, we have the following itive root. For a2 < 0, f ′(IV∗ ) has two negative real roots. f (0) a0 < 0 R0 < 1 f (I ) relations Since = for , then, V∗ has no positive real roots, and consequently no endemic equilibria. µ θ (µ θ )S θ I H m1β S I , h + h = h + h H∗ + h ∗ + h H∗ V∗ SH∗ IV∗ Furthermore, we can use the Descartes’ Rule of Signs µh γh m1βh , + = I∗H [7] to explore the existence of endemic equilibrium (or IF∗ U∗ R 1 m2βv 1 IV∗ ηvβv 1 IV∗ , equilibria) for 0 < 1. We note the possible existence of = − I∗V + − I∗V I∗V U∗ µf m3βf 1 IF∗ ηf βf 1 IF∗ , backward bifurcation. The theorem below summarises = − IF∗ + − IF∗ IF∗ IV∗ the existence of endemic equilibria of the model system µe m4σf m5σv . (11) U∗ U∗ (2). = + Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 8 of 15

Evaluating the components of the time derivative of the so as to make the coefficients ofx, y and z equal to zero. Lyapunov function using the relations (11) we have We have S S I S I V H∗ (µ θ ) H θ H β H V ˙ 1 h h SH∗ 1 hIH∗ 1 m1 hSH∗ IV∗ 1 = − SH + − SH∗ + − IH∗ + − SH∗ IV∗ IH∗ SH IV IH IV∗ IF IV A 1 m1βhSH∗ IV∗ B 1 m2βvIF∗ + − IH SH∗ IV∗ − IH∗ + − IV IF∗ − IV∗ IF U IV U m2βvIF∗IV 1 ηvβvU ∗ ηvβvU ∗IV 1 + − IF∗ + U ∗ − IV∗ + − U ∗ (12) IF∗ U IF U C 1 ηf βf U ∗ ηf βf U ∗IF 1 + − IF U ∗ − IF∗ + − U ∗ IV IF IV m3βf IV∗ m3βf IV∗ IF 1 + IV∗ − IF∗ + − IV∗ U ∗ IF U IV U D 1 m4σf IF ∗ m5σvIV ∗ . + − U IF∗ − U ∗ + IV∗ − U ∗

Let m1βhSH∗ IV∗ SH IH IV IF U A 1, B . , , , and . v w x y z (13) = = m2βvIV∗ (1 IF∗) ηβvU ∗(1 IV∗ ) = SH∗ = IH∗ = IV∗ = IF∗ = U ∗ − + − The coefficients C and D can similarly be evaluated from Substituting (13) into (12), we obtain the coefficients of y and z. Note that expressions such as 2 (1 v) 1 xv V (µh θh)S∗ − H(v, w, x, y, z), (14) ˙ H m1βhS∗ I∗ 2 =− + v + H V − v − w where 1 w 1 H θhI∗ 1 w m1βhS∗ I∗ 1 x xv = H − − v + v + H V − v + − vx x Am1βhS∗ I∗ 1 xv w Bm2βvI∗ 1 y x + H V + − − w + F + − − y z Bm2βvI∗I∗V x y xy 1 BηvβvU ∗ 1 z x + F + − − + + − − x x BηvβvU ∗I∗V (x z xz 1) Cm3βf IV ∗ 1 x y + + − − + + − − y (15) z Cm3βf IV ∗IF ∗ y x xy 1 Cηf βf U ∗ 1 z y + + − − + + − − y y Cηf βf U ∗IF ∗ y z yz 1 Dm4σf IF ∗ 1 y z + + − − + + − − z x Dm5σvIV ∗ 1 x z . + + − − z

Next, we choose A, B, C and D so that none of the vari- emanating from the substitution of the coefficients into H H able terms of are positive. It is important to group , are less than or equal to zero by the arithmetic mean- H H together the terms in that involve the same state vari- geometric mean inequality. This implies that 0 with SH IH IV IF U 1.≤ able terms, as well as grouping all of the constant terms equality only if S IH IV I U H < H ∗ = ∗ = ∗ = F ∗ = ∗ = together. So we can show that 0 by expanding (15), V Therefore, 0 and by the LaSalle’s Extension [17], it writing out the constant term and the coefficients of the ˙ ≤ 1 w x implies that the omega limit set of each solution lies in variable terms such as v, w, x, y, z, v , v , v and so on. The an invariant set contained in . The only invariant set only variable terms that appear with positive coefficients E contained in is the singleton 1. This shows that each are x, y and z. We thus choose the Lyapunov coefficients Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 9 of 15

R5 solution which intersects limits to the endemic equi- Table 2 Parameter values used for the simulations + and sensitivity analysis librium. This completes the proof. Parameter Value/range Source Results 5 µh 4.5 10 Parameter estimation − [21] × 5 γh 1.6 10− 0.5 [23] The biggest challenge in epidemic modeling is the esti- × − 2 θh 0 1.1 10 [23] mation of parameters in the model validation process. In − × − m , m m 1, m 1 this section we endeavour to estimate some of the param- 1 2 1 < 2 > Estimated m , m , m eter values of system (2). The demographic parameters 3 4 5 (0,1) Estimated , , can be easily estimated from census population data. We βh βf βv (0,1) Estimated begin by estimating the mortality rate µh. We note that ηv (1,5) Estimated the average life expectancy of the human population in ηf (0, 1) Estimated , Ghana is 60 years [21]. This translates intoµ h 0.017 σf σv (0,1) Estimated 5 = 3 3 4.6 10− µf 3 10− 7 10− Estimated per year or equivalently × per day. Buruli ulcer × − × is currently regarded as a vector borne disease. Recovery µe (0,1) Estimated rates modelled by γh, of vector borne diseases range from 1.6 10 5 × − to 0.5 per day [23]. This translates to an average of between 0.00584 and 183 per year. The rate of loss each of the parameters. If the magnitude of the PRCC of immunity θh for vector borne diseases range between value of a parameter is greater than 0.5 or less than 0.5 1.1 10 2 − 0 and × − per day[23]. The mortality rate of the and the p-value less than 0.05, then the model is sensitive water bugs is assumed to be 0.15 per day [3]. The rates to the parameter. On the other hand, PRCC values close 1 1 per day can easily be transferred to yearly rates. to + or − indicate that the parameter strongly influ- In this model we shall assume that we have more water ences the state variable output. The sign of a PRCC value bugs than humans so that m1 < 1. Since the water bugs indicates the qualitative relationship between the param- prey on the fish, a reasonable food chain structure leads eter and the output variable. A negative sign indicates to the assumption that we have more fish than water that the parameter is inversely proportional to the out- bugs hence m2 > 1 and consequently 0 < m3 < 1. If the come measure [10]. The parameters with negative PRCCs water bug is assumed to interact more with the environ- reduce the severity of Burili ulcer disease while those ment than fish thenη v > 1 and 0 <ηf < 1. The natural with positive PRCCs aggravate it. Using Latin Hyper- mortality of small fish in rivers is not well documented cube Sampling (LHS) scheme with 1000 simulations for and data on the mortality of river fish in Ghana is not each run, with U as the outcome variable. Our results available. For the purpose of our simulations, we shall show that the variable U is sensitive to the changes in the 3 10 3 <µ < 7 10 3 m , η , µ , µ β assume that × − f × − per day. Given parameters 3 f e f and f . The results are shown K NF , NV 0 m4, m5 1. that ≥ we have ≤ ≤ We shall also in Fig. 2. 0 σ , σv 1. assume that ≤ f ≤ We summarise the param- The results from the PRCC analysis are summarized in eters in the following Table 2. Table 3. The significant parameters together with their PRCC values and p-values have been encircled. Sensitivity analysis In Fig. 3 the residuals for the ranked Latin Hypercube Many of the parameters used in this paper are not deter- Sampling parameter values are plotted against the residu- mined experimentally. Therefore their accuracy is always als for the ranked density of Mycobacterium ulcerans. in doubt. This can be overcome by observing responses The PRCC plots for parametersβ f , µf , µe and ηf show a of such parameters and their influence on the model strong linear correlation. The growth of Mycobacterium variables through sensitivity and uncertainty analysis. ulcerans increases as the number of infected fish that In this subsection we present the sensitivity analysis of eventually shed bacteria into the environment increases. the model parameters to ascertain the degree to which An increase in the parameters µf and µe leads a decrease the parameters affect the outputs of the model. We use in amount of bacteria in the environment. the Partial Correlation Coefficients (PRCCs) analysis to determine the sensitivity of our model to each of the Data and the fitting process parameters used in the model. Through correlations, the One of the most important steps in the model building association of the parameters and state variables can be chronology is model validation. We now focus on the data established. In our case, we determine the correlation of provided by the Ashanti Regional Disease Control Office for our parameters and the state variable U. Alongside the Buruli ulcer cases in Ghana per 10,0000 people. The data are PRCCs are the statistical significance test p-values for given in the Table 4 below for the years 2003–2012. Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 10 of 15

m 3

ηf

µe

µf

βf

µv

ηv

βv m 2

γh

βh

m 1

θh

µh −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Fig. 2 PRCC plots: The variable U largely depends on m3, ηf , µe, µf and βf. The bars pointing to the left indicate that U has an inverse dependence on the respective parameters. We observe that the parameters m3, ηf and βf aggravate the disease when they are increased while µf and µe reduce its severity when increased

Table 3 Outputs from PRCC analysis Parameter PRCC p-value Parameter PRCC p-value

µ 0.0278955496902069 0.15143 ηv -0.0115013733722167 0.6938 h − θh 0.00442843658688886 0.9812 µv -0.0180489104210923 0.46911

m1 0.0515998428094382 0.80091 β 0.778665515455008 1.8606e 207 f − β -0.0157345069415098 0.86872 µ -0.700364635558767 1.3865e 159 h f −

γ -0.0107900263078278 0.44864 µe -0.730147310381517 3.8349e 165 h −

m2 0.0234088416548564 0.87524 η 0.671003774168464 3.5184e 145 f −

βv 0.0241407300242573 0.10143 m3 0.521170177172971 4.476e 067 −

We fit the model system (2) to the data of Buruli ulcer squares-curve fitting method. Matlab code is used where cases expressed as fractions. We use the least squares unknown parameter values are given a lower and upper curve fit routine (lsqcurvefit) in Matlab with optimisation bound from which the set of parameter values that pro- to estimate the parameter values. Many parameters are duce the best fit are obtained. known to lie within limits. A few parameters such as the Figure 4 shows how system (2) fits to the available data demographic parameters are known [13] and it is thus on the incidence of the BU. The incidence solution curve important to estimate the others. The process of esti- shows a very reasonable fit to the data. mating the parameters aims at finding the best concord- In planning for a long term response to the Buruli ulcer ance between computed and observed data. One tedious epidemic, it is important to have some reasonable pro- way to do it is by trial and error or by the use of software jections to the epidemic. The fitting process allows us to programs designed to find parameters that give the best envisage the Buruli ulcer epidemic in future. it is impor- fit. Here, the fitting process involves the use of the least tant to note that the projections are reasonably good over Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 11 of 15

[PRCC , p−value] = [10 , 2.7608e−204]. [PRCC , p−value] = [11 , 2.4106e−148]. 600 600 µ βF F

400 400

200 200

U 0 U 0

−200 −200

−400 −400

−600 −600 −800 −600 −400 −200 0 200 400 600 −500 0 500 µ βF F

[PRCC , p−value] = [13 , 4.6405e−167]. [PRCC , p−value] = [15 , 7.9046e−132]. 600 600 η µE f

400 400

200 200 U U 0 0

−200 −200

−400 −400

−600 −600 −600 −400 −200 0 200 400 600 800 −600 −400 −200 0 200 400 600 η µE f

[PRCC , p−value] = [16 , 9.3899e−071]. 600 m 3

400

200

U 0

−200

−400

−600 −500 0 500 m 3 Fig. 3 PRCC plots shows the PRCC plots for the parameters βf, µf, µe, ηf and m3 Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 12 of 15

Table 4 Data on Buruli ulcer cases in Ghana

Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Buruli ulcer cases 739 1159 1201 1096 1136 1300 1158 1428 1324 1292

Source [13]

0.2 Model fit Actual data 0.18

0.16

0.14

0.12

0.1

0.08 BU Incidence

0.06

0.04

0.02

0 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Time in years Fig. 4 Model fit to data. Model system (2) fitted to data of Burili ulcer cases in Ghana. The circles indicate the actual data and the solid line indicates the model fit to the data. The parameter values used for the fitting;µ h = 0.000045, θ = 0.1, m1 = 5, βh = 0.1, γ = 0.056, m2 = 10, βv = 0.000065, ηv = 1.5, ηf = 0.6, µV = 0.15, βf = 0.00005, µf = 0.05, σf = 0.05, σv = 0.006, µe = 0.4

a short period of time since the current is evolving gradu- humans. Increasing µe leads to a decrease in the preva- ally based on the available data. We chose to project the lence of infected humans. epidemic beyond 5 years to 2017. Figure 5 show the projected Buruli ulcer epidemic. Discussion Figures 6 and 7 show the changes in the prevalence of In this paper, a deterministic model on the dynamics of infected humans respectively when σf , the shedding rate the Buruli ulcer in the presence of a preventive interven- of Mycobacterium ulcerans in the environment and µe tion strategy is presented. The model’s steady states are the removal rate of MU from the environment, are var- determined and their stabilities investigated in terms of R ied. Based on the sensitivity analysis, our model is very the classic threshold 0. In disease transmission mod- sensitive to the shedding rate of Mycobacterium ulcerans elling, it is well known that a classical necessary condi- into the environment. Figure 6 shows that an increase tion for disease eradication is that the basic reproductive R in the shedding rate will lead increased human infec- number 0, must be less than unity. The model has mul- tions. We can actually quantify the related increases. tiple endemic equilibria (in fact it exhibits a backward For instance, if σf is increased from 0.51 to 0.52 on year bifurcation). When a backward bifurcation occurs, 15, the percentage increase in the prevalence of human endemic equilibria coexist with the disease free equi- R infections is 6 %. Minimising Mycobacterium ulceransin librium for 0 < 1. This means that getting the classic R the environment is an important control measure that is, threshold 0 less than 1, might not be sufficient to elimi- albeit impractical at the moment. We observe through nate the disease. Thus the existence of backward bifur- our results that their decrease in the environment can cation has important public health implications. This lead to quantifiable changes in the prevalence of infected might explain why the disease has persisted in the human Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 13 of 15

0.2 Model fit Actual data

0.18

0.16

0.14

0.12

0.1 BU Incidence 0.08

0.06

0.04

0.02

0 2004 2006 2008 2010 2012 2014 2016 2018 Time in years Fig. 5 Projected model fit. Projection to fit in Fig. 4

0.016

0.015

0.014

0.013

0.012

0.011

0.01 Prevalence of infected humans

0.009 =0.54 σf =0.53 σf 0.008 =0.52 σf =0.51 σf 0.007 0 5 10 15 20 25 30 Time(t) in years Fig. 6 Prevalence of Buruli ulcer infection in humans. Shows prevalence humans when σf is varied Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 14 of 15

0.14

0.12

0.1

0.08

0.06 Infected Humans

0.04

0.02 =0.015 µe =0.02 µe =0.025 µe =0.03 µe 0 0 5.5 11 16.6 22 27.5 30 Time (years) Fig. 7 Prevalence of Buruli ulcer in infected humans for different values ofµ e. Shows prevalence the infected humans when µe is varied

population over time. The endemic equilibrium is found necessary and very important to determine how these R to be globally stable if 0 > 1. parameters influence the model. The implications of The sensitivity analysis of model parameters showed varying some of the important epidemiological param- some interesting results. These results suggest that efforts eters such as the shedding rates were investigated. to remove Mycobacterium ulcerans and infected fish Important results were drawn through Figs. 6 and 7. from the environment will greatly reduce the epidemic The main result of this paper is that the management of although the latter will be impracticable. This is because Buruli ulcer depends mostly on the management of the of the costs involved and the fact that many governments environment. in affected areas operate on lean budgets. The model is then fitted to data on the Buruli ulcer Conclusions in Ghana. The model reasonably fits the data. The chal- This model can be improved by considering social inter- lenge in the fitting process was that the data appears ventions in the human population, modeled as functions to indicate that Buruli ulcer has reached a steady and the inclusion of the different forms of treatment state. This then produced some parameter values that available as some individuals opt for traditional methods appeared unreasonable. Despite these challenges, the while others depend on the government health care sys- fit produced reasonable projections on the future of the tem [1]. Social interventions include education, aware- ulcer. The model shows that in the near future, the num- ness, poverty reduction and provision of social services. ber of cases will not change if everything remains the While the mathematical representations of these inter- same. An important consideration that can be added to ventions are insurmountable, they are vital to the dynam- the model is the inclusion of probable policy shifts and ics of the disease and public health policy designs. Finally the investigation of different scenarios on the progres- this model can be used to suggest the type of data that sion of the epidemic as the policies change. Because not should be collected as research on the Buruli ulcer inten- much of the disease is understood, parameter estima- sifies. The global burden of the disease and its epidemiol- tion was a daunting task. So we had to reasonably esti- ogy are not well understood, [28]. Clearly, gaps do exist in mate some of the parameter using the hypothesis that the nature and type of data available. Reports on the dis- Buruli ulcer is a vector borne disease. Due to the esti- ease are often based on passive presentations of patients mation of essential parameters sensitivity analysis was at health care facilities. As a result of the difficulties of Nyabadza and Bonyah BMC Res Notes (2015) 8:656 Page 15 of 15

accessing health care in affected areas, data on the dis- 10. Gomero B. Latin Hypercube Sampling and Partial Rank Correlation Coef- ficient analysis applied to an optimal control problem, MSc Thesis, The ease is scanty. University of Tennessee. 2012. 11. Grassly NC, Fraser C. Mathematical models of infectious disease transmis- Authors’ contributions sion. Nat Rev Microbiol. 2008;6:477–87. FN designed the model and carried out the numerical simulations. EB did the 12. Grundmann H, Hellriegel B. Mathematical modelling: a tool for hospital mathematical analysis and writing of the manuscript. Both authors read and infection control. Lancet Infect Dis. 2005;6(1):39–45. approved the final manuscript. 13. Ghana Statistical Service, Available at: http://www.statsghana.gov.gh. Accessed Sept 2013. Author details 14. Houben RM, Dowdy DW, Vassall A, et al. How can mathematical models 1 Department of Mathematical Sciences, Stellenbosch University, Private Bag advance tuberculosis control in high HIV prevalence settings? Int J Tuber X1, Matieland 7602, South Africa. 2 Department of Mathematics and Statistics, Lung Dis. 2014;18(5):509–14. Kumasi Polytechnic, P. O. Box 854, Kumasi, Ghana. 15. Huppert A, Katriel G. Mathematical modelling and prediction in infectious disease epidemiology. Clin Microbiol Infect. 2013;19:999–1005. Acknowledgements 16. Jacobsen KH, Padgett JJ. Risk factors for Mycobacterium ulcerans infec- The first author acknowledges with gratitude the support from the Stellen- tion. Int J Infect Dis. 2010;14(8):e677–81. bosch University, International office for the research visit that culminated into 17. LaSalle JP. The stability of dynamical systems. In: CBMS-NSF Regional this manuscript. The second author acknowledge, with thanks, the support of Conference Series in Applied Mathematics 25, SIAM: Philadelphia. 1976. the Department of Mathematics and Statistics, Kumasi Polytechnic. 18. Marsollier L, Robert R, Aubry J, Andre JS, Kouakou H, Legras P, Manceau A, Mahaza C, Carbonnelle B. Aquatic insects as a vector for Mycobacterium Competing interests ulcerans. Appl Environ Microbiol. 2002;68:4623–8. The authors declare that they have no competing interests. 19. Marty R, Roze S, Bresse X, Largeron N, Smith-Palmer J. Estimating the clini- cal benefits of vaccinating boys and girls against HPV-related diseases in Received: 20 May 2015 Accepted: 26 October 2015 Europe. BMC Cancer. 2013;19:19. doi:10.1186/1471-2407-13-10. 20. Merritt RW, Walker ED, Small PLC, Wallace JR, Johnson PDR, et al. Ecol- ogy and transmission of buruli ulcer disease: a systematic review. PLoS Neglect Trop Dis. 2010;4(12):e911. 21. Population and Housing Census National Analytical Report, 2012. http:// References www.statsghana.gov.gh. 1. Agbenorku P, Donwi IK, Kuadzi P, Saunderson P. Buruli Ulcer: treat- 22. Portaels F, Chemlal K, Elsen P, Johnson PD, Hayman JA, Hibble J, Kirkwood ment challenges at three centres in Ghana. J Trop Med. 2012; R, Meyers WM. Mycobacterium ulcerans in wild animals. Revue Scienti- doi:10.1155/2012/371915. fique et Technique. 2001;20:252–64. 2. Ahorlu CK, Koka E, Yeboah-Manu D, Lamptey I, Ampadu E. Enhancing 23. Rascalou G, Pontier D, Menu F, Gourbie‘re S. Emergence and prevalence Buruli ulcer control in Ghana through social interventions: a case study of human vector-borne diseases in sink vector populations. PLoS One. from the Obom sub-district. BMC Public Health. 2013;13:59. 2012;7(5):e36858. doi:10.1371/journal.pone.0036858. 3. Aidoo AY, OSei B. Prevalence of acquatic insects and arsenic concentra- 24. Silva MT, Portaels F, Pedrosa J. Aquatic insects and Mycobacterium tion determine the geographical distribution of Mycobacterium ulcerans ulcerans: an association relevant to buruli ulcer control? PLoS Med. infection. Comput Math Method Med. 2007;8:235–44. 2007;4(2):e63. 4. Boleira M, Lupi O, Lehman L, Asiedu KB, Kiszewski Ana Elisa. Buruli ulcer. 25. Sopoh GE, Barogui YT, Johnson RC, Dossou AD, Makoutode M. Family Anais Brasileiros de Dermatologia. 2010;85(3):281–301. relationship, water contact and occurrence of Buruli ulcer in Benin. PLoS 5. Clift E. IEC interventions for health: a 20 year retrospective on dichoto- Neglect Trop Dis. 2010;4(7):e746. mies and directions. Int J Health Commun. 1998;3(4):367–75. 26. Stienstra Y, van der Graaf WTA, Asamoa K, van der Werf TS. Beliefs and atti- 6. Debacker M, Portaels F, Aguiar J, Steunou C, Zinsou C, Meyers W, et al. Risk tudes toward buruli ulcer in Ghana. Am J Trop Med Hyg. 2002;67:207–13. factors for Buruli ulcer, Benin. Emerg Infect Dis. 2006;12:1325–31. 27. Williamson HR, Benbow ME, Campbell LP, Johnson CR, Sopoh G, Barogui 7. Descartes’ Rule of Signs, Available at: http://www.purplemath.com/ Y, Merritt RW, Small PLC. Detection of Mycobacterium ulcerans in the modules/drofsign.htm, Acessed 2 Sept 2013. environment predicts prevalence of buruli ulcer in Benin, PLoS Neglect 8. Eddyani M, Ofori-Adjei D, Teugels G, De Weirdt D, Boakye D, Meyers WM, Trop Dis. 2012; e1506. Portaels F. Potential role for fish in transmission of Mycobacterium ulcerans 28. World Health Organization. Buruli ulcer Mycobacterium ulcerans infection, disease (Buruli Ulcer): an environmental study. Appl Environ Microbiol. http://www.who.int/buruli/en/. 2004;5679–81. 29. World Health Organization. Buruli ulcer: Number of new cases of Buruli 9. Garba SM, Gumel AB, Abu Bakar MR. Backward bifurcaytion in dengue ulcer reported (per year). http://apps.who.int/neglected_diseases/ntd- transmission dynamics. Math Biosci. 2008;215:11–25. data/buruli/buruli.html.

Submit your next manuscript to BioMed Central and take full advantage of:

• Convenient online submission • Thorough peer review • No space constraints or color ﬁgure charges • Immediate publication on acceptance • Inclusion in PubMed, CAS, Scopus and Google Scholar • Research which is freely available for redistribution

Submit your manuscript at www.biomedcentral.com/submit