Modelling Transmission of Buruli Ulcer in the Central Region of Ghana

: The pathogen of Buruli Ulcer (BU) is known to be Mycobacterium Ulcerans whose mode of transmission is entirely not known, although the disease is recognised to be associated with contaminated water. The hypothesised transmission involves humans being bitten by the water bugs (vector) that prey on mollusks, snails and young ﬁshes. The hypothesised transmission also involves humans feeding on an infected fish or frog. This study seeks to contribute to the dynamics and analyses of the transmission mechanism of Buruli Ulcer in communities along Offin River in the Central Region of Ghana. The model equilibria were determined and conditions for the equilibria were also established. The basic reproduction number, 0 R was derived using the Next Generation approach and its estimated value was 1.20771. The result reveals that, 0 R is greater than 1, indicating a horizontal spread of the infection across the population. The transmission dynamics of Buruli Ulcer model of the Susceptible, Infected and Recovered (SIR) type also show that the disease will continue to spread at the study areas as long as the reservoir for Mycobacterium Ulcerans continue to sustain enough infected water bugs and infected fish or infected frog to contain the disease. The study further concludes that, the rate of spread of Buruli Ulcer in the affected communities continue to be high due to its mode of transmission. This study suggest that adequate control measures including mass education and prompt treatment to curb the spread should be emphasized.


Introduction
Modelling of infectious disease is a tool which has been used to study the means by which diseases spread, forecast the future course of an outbreak and to evaluate strategies to control an epidemic [1]. Mathematical models may not offer comprehensive descriptions of how to control diseases. They are elegant methods for evaluating the possible influence of different strategies offered in public health intervention programs [2].
Buruli Ulcer (BU) is a disease of the skin caused by Mycobacterium Ulcerans, a slow growing mycobacterium that classically infect the skin and subcutaneous tissue, giving rise to indolent no ulcerated (nodules, plaques) and ulcerated lesions. After tuberculosis and leprosy, Buruli Ulcer is the third most common mycobacteriosis of human. The mode of Mycobacterium Ulcerans transmission is not fully understood in the study area, although the disease is recognized to be associated to contaminated water. Areas affected by Buruli Ulcer disease are located near stagnant or slow-moving water, and outbreaks appear to be related to environmental changes (deforestation, agriculture, hydraulic installations) involving surface water [3].
The occurrence of Buruli Ulcer upstream of the Offin River is greater and steeply dwindles as you go beyond Dunkwa. So, the communities on the upstream of river Offin from Agroyesum (e.g. Subin, Ameyaw, Betenase, Ampabena, Nkotumso, Dominase, Ayanfuri, Powerline, Obiarabiaradanedanmu, Pokukrom and Dunkwa) are very endemic [3]. Therefore, to provide awareness of this endemic, this paper seeks to develop a transmission model (deterministic model) of Buruli Ulcer and help to identify control measures that will minimize the disease in the study area. The communities along Offin River in the Central Region of Ghana are endemic to Buruli Ulcer disease. Figure 1 present the communities along Offin River. These areas of study are located near stagnant or slow-moving water, and experience similar environmental changes (deforestation, agriculture, and mining) involving surface water. The primary means of securing the necessities of life in these communities are farming and mining [3]. It can be seen from the Powerline community ( Figure 2), most of the cases are in Zones 3 and 4 and also at the fringes of the community. Communities such as Betenase, Ampabena ( Figure 1) have Buruli Ulcer cases all over the community which seems to portray that the whole community is endemic. Figure 3 portrays that Zone 1 of Ayamfuri has about 70% of Buruli Ulcer cases and even in that zone, 90% were clustered together in a small sector within the centre of the town. Scot proposed a mathematical model of the SIR-type in an endeavour to explain the role of aquatic insects and arsenic in the spread of Buruli ulcer disease [6]. The paper considered arsenic environment as a reservoir for Mycobacterium ulcerans and water bug (vector) for Buruli Ulcer disease. In their model, they proposed that Buruli Ulcer is a micro parasitic disease in which host parasite interaction basically occurs within isolated communities. Again, it was assumed that the host population is of fixed size containing susceptible individuals who are not yet infected with Mycobacterium Ulcerans. It also assumed humans who develop Buruli Ulcer become immune to any further attack and this assumption led them to the SIR model. The model equations describing the proportion of humans infected by Mycobacterium Ulcerans and the corresponding proportion of water-bugs according to them are given by:  The study displayed a nonlinear relationship between the basic reproductive number 0 ( ) R and the widespread of infection of both infected water bugs and infected humans at a particular time. However, a small increase in the reproductive number lead to a large change in both the prevalence levels of humans and water bugs. It was deduce from their graph that, higher levels of 0 ( ) R will lead to increases in cases of Buruli Ulcer for 0 1 R > . Thus, if Buruli Ulcer is not controlled it will continue to spread in regions with unconducive conditions. Nyabadza and Bonyah develop the transmission dynamics of Buruli Ulcer in Ghana: Insights through a mathematical model [7]. In their paper, a model for the transmission of Mycobacterium Ulcerans to human in the presence of a preventive strategy was proposed and analysed. The model equilibria were determined and conditions for the existence of the equilibria established. The model analysis was carried out in terms of the reproduction number 0 R . The disease-free equilibrium was found to be locally asymptotically stable for 0 1 R < . The model was then fitted to data from Ghana. The dynamics of the disease was described by the following set of nonlinear set differential equations;  The model exhibits a backward bifurcation and the endemic equilibrium point is globally stable when 0 1 R < . Sensitivity analysis also showed that the Buruli Ulcer epidemic is highly influenced by the shedding and clearance rates of Mycobacterium Ulcerans in the environment. The model was found to fit reasonably well to data from Ghana and projections on the future of the Buruli Ulcer epidemic were also made. Mycobacterium Ulcerans are transferred from vector (water bug), fish and frog to the humans.

Methodology
There is homogeneity of human, vector and fish-frog population's interactions.
Infected humans recover and are temporarily immune, but eventually lose immunity.
Fish and amphibian (frog) are preyed on by the vector and the human.
The acquiring of Mycobacterium Ulcerans through environmental contact and direct person-to-person transmission is rare [8].
Susceptible host are more likely to be infected by the disease through biting by an infectious vector (water bug). Being the incidence of new infections transmitted to a human when the human prayed on an infected fish or frog.
Susceptible insect is infected at a rate ( ) through the predation of infected fish or frog and ( ) representing other sources in the environment. Where η differentiates the infectivity potential of the fish-frog from that of the environment.
The vector population and the fish-frog population are assumed to be constant. Their growth functions are given by ( ) V g P and ( ) FF g P respectively. Generally, we can assume that ( ) There is a proposed hypothesis that environmental mycobacteria in the bottoms of swamps may mechanically concentrated by small water-filtering organisms such as microphagous fish, snails, mosquito larvae, small crustaceans and protozoa [9]. It can therefore be assumed that fish-frog increase the environmental concentrations of mycobacterium Ulcerans at a rate FF σ . Humans are assumed not to shed any bacteria into the environment. Vector release bacteria into the environment at a rate V σ .
The model does not include a potential route of direct contact with the bacterium in water.
The birth rate of the human population is directly proportional to the size of the human population.

Identification of the Biological Interest of the Model
The biological region of interest of Equation (3) where H P is constant. The above model (3.6) can be solved by equating the righthand side of the model to zero. This computation shows that the model always has a disease free equilibrium (DFE) at Evaluating T at the disease-free equilibrium also gives Equation (21) 0 0 Let ∑ be the transition state of Equation (3).
Hence the transition state of the disease can be expressed as Equation (22) Taking the Jacobian matrix of 1 Hence the inverse of Equation (24) is computed as Equation (25) 1 1 0 0 0 ( ) Let L K = Next Generation Matrix (NGM) with large domain. Therefore, the NGM is computed as Equation (26) 1 Simplifying Equation (27) Let n λ be the eigenvalues of L K . Then the eigenvalues of Equation (28) is computed as Equation (29) det( ) 0 Where I is a 4 4 × identity matrix and 1, 2,3, 4 n = . Hence the eigenvalues of Equation (28) are obtained as Equation (30) The basic reproductive number ( 0 R ) is the spectral radius of L K which is written as Equation (31) 0 ( ) Hence the basic reproduction number is largest eigenvalue of Equation (30). Therefore Hence the basic reproduction number is given by Equation The basic reproduction number of the model is determined by the vector population, the density of Mycobacterium Ulcerans in the environment and the infected Fish-Frog population from that environment. The 0 R actually appears to be influenced by the rate at which Mycobacterium Ulcerans is introduced into the environment and the infectivity potential of the Fish-Frog from that of the environment.

The Stability of the Disease-Free Equilibrium (DFE)
Let 0 E J be the Jacobian evaluated at DFE of the model, therefore Simplifying Equation (36), 0 R is obtained as Equation (37) below From Equation (37), it implies that 7 λ is negative.
Since the eigenvalues of 0 E J are all negatives it therefore shows that the disease-free equilibrium is locally asymptotically stable for 0 1 R < .

The Stability of the Endemic Equilibrium
The endemic equilibrium ee E of the model is locally asymptotically stable if and only if 0 1.
R > However it is very difficult to deal with the stability of the endemic equilibrium analytically due to the nature of transmission model (i.e. Model 3). By numerical approach, the endemic equilibrium is locally asymptotically stable. Different initial conditions were applied for the simulation. Those obits shown to be the same point as time evolve.

Results and Discussion
Numerical simulation of the Equation (3.6) was carried out using Maple inbuilt functions. Analysis of the response of model parameters on the transmission dynamics of the disease was studied and the evaluation of the basic reproduction number was made. Since, most of the parametric values are not readily available it is needed to assign some arbitrary values. However, some are available at [6,11,12]. The initial conditions were taken at initial time of zero (0) and the final time was considered as 500, 500, 500, 500, 520, 2000, 2000, 2000, 7000, 8000 and finally 10000. The results of the simulation study are presented in Figure 8. From Figure 7 the model simulations show that the susceptible human population decrease and becomes steady with time. The infected human population also increases so sharply and becomes asymptotic with time. This is due to the rate at which a susceptible human is bitten by an infected water bug and also the rate at which a susceptible human feed on an infected fish-frog. It can be observed that the Recovered human population increases to a certain point and it then decreases asymptotically with time.  Figure 8 shows the general interactions between human and vector population in the study areas with time. It's also shows that there will be a widespread occurrence of the Buruli Ulcer disease in study areas.
The basic reproduction number was calculated as Equation Chapter (Next) Section 1 (38) The positivity of solution to the model was proved as shown in Equation (4) to (17). The Basic Reproduction Number 0 ( ) R was computed from the deterministic model that was developed in Equation (4). The model's equilibria were determined and conditions for the equilibria were also established, and their stabilities were investigated in terms of the classic threshold 0 ( ) R . It is well known in disease transmission modelling that, a classical necessary condition for minimizing or eliminating disease completely is that the basic reproduction number 0 ( ) R , must be less than unity. If the basic reproduction number is greater than unity, then the disease will eventually spread in a population. The diseasefree equilibrium (DFE) is found to be locally asymptotically stable for 0 1 R < as it's shown in Equation (37). The endemic equilibrium is also found to be locally asymptotically stable for 0 1 R > . A simulation was performed on the model using some of the parameters in Table 3. The results obtained indicate that, the disease has attained a steady state, this is typically shown in Figure 8. In examining the state of Buruli Ulcer in the study area, since 0 1 R > it proves that the number of infected persons in the communities will increase with time.

Conclusions
The study developed a deterministic model for Buruli Ulcer based on the transmission mechanism of the disease in the study area. The basic reproduction number 0 R derived was found to be greater than unity (i.e. 0 1 R > ), this shows that, the disease will spread horizontally across the population, and the estimated value of 0 R was 1.20771. The transmission dynamics of Buruli Ulcer model of the Susceptible, Infected and Recovered (SIR) type showed that, the disease will continue to spread at the study areas as long as the reservoir for Mycobacterium Ulcerans continue to increase and also as long as enough infected water bugs and infected fish or infected frog continue to increase. However, the spread of the disease (Buruli Ulcer) will be minimised when control measures are being implemented. It is therefore recommended that People in the affected communities should be given adequate education on the Buruli Ulcer disease, how contaminated water or stagnant water play a major role in the transmission of the disease and how the disease should be treated properly to minimize its spread.