Mathematical models constructed for epidemic diseases are to study the nonlinear process involved in the dynamics of infectious diseases and to determine the best possible way for its control. Among other infectious diseases, one can cite the Buruli ulcer (BU). The World Health Organization (WHO) defines Buruli ulcer (BU) as “an infectious disease involving the skin, caused by Mycobacterium ulcerans, characterized by a painless nodule, papule, plaque or edema, evolving into a painless ulcer with undermined edges, often leading to invalidating sequelae. Sometimes bones are destroyed” [1] . After tuberculosis and leprosy, BU is the third most common mycobacterial disease. Incidences have increased recently, especially in West Africa [2] [3] .

In 1997, WHO recognized BU as an emerging public health problem. In West Africa countries, the prevalence of Buruli ulcer has been increasing in recent years [1] [3] . Buruli ulcer-endemic foci are regularly associated with stagnant bodies of water (ponds, backwaters, and swamps). The disease takes various clinical forms, including ulcers, nodules, plaques, and oedematous indurations. Surgical excision followed by skin grafting is the recommended treatment [4] . M. ulcerans is an environmental mycobacterium associated with wetlands, especially slow-flowing or stagnant water [5] [6] . Aquatic insects may play a role in transmitting BU to humans [7] [8] . Naturally, acquired M. ulcerans infection in wild animals [9] suggests that the etiologic agent is an environmental organism. Bone lesions also exist [10] . Trauma is probably the most frequent means by which M. ulcerans is introduced deep into the skin or subcutaneous tissue [11] , where a toxin is produced, causing massive necrosis of the skin. Buruli ulcer is therefore clinically primarily a disease of the skin. M. ulcerans also cause osteomyelitis: in Benin, patients have been reported to suffer from severe forms of osteomyelitis as a consequence of an M. ulcerans infection, sometimes necessitating amputation [12] .

The mode of transmission of the disease in humans is attributed to an environmental bacterium Mycobacterian ulcerans. However, this mode of transmission, remains uncertain because of the mysterious character of the bacterium. One possibility is that the disease is transmitted to humans by certain insects found in water. Nowadays, there is no established correlation between infection of humans and animals. For the treatment of the most serious cases, hospitalization is necessary. Indeed, hospitalization is generally prolonged thereafter, reaching on average 3 months per patient, but can reach 18 months or more. Antibiotic treatment is unfortunately unsatisfactory.

Several mathematical models are developed in order to better understand the evolution of Buruli ulcer within a human population. In [13] , the authors formulate a system governed by nonlinear mathematical equations. Analytical results show that the elimination of certain taxa, notably Oligochaeta worms, can effectively reduce mycobacterial transmission in aquatic ecosystems, unlike the elimination of the most abundant taxa. In [14] , a mathematical model of Buruli ulcer with a saturated processing function is developed with the aim of theoretically modeling the possible impact of challenges associated with the treatment and management of BU, such as delays in accessing treatment, limited resources and few medical facilities to deal with the very complex treatment of ulcer. Other authors such as [15] propose a mathematical model of Buruli ulcer transmission integrating optimal control strategies. [16] proposes a model of leprosy, a disease of the same category as UB.

Despite the numerous efforts already made, Buruli ulcer remains a public health issue in several countries, including Benin. Diagnosed patients are hospitalized for the entire duration of their treatment.

However, how do environmental factors and individuals’ loss of follow-up influence the understanding of Buruli ulcer transmission dynamics, and what are the potential impacts of this integration on disease prevention and control strategies?

This work introduces a new mathematical model and intervention strategies that integrate the environment as well as individuals lost to follow-up (those who, due to lack of resources or for cultural reasons, do not return to the hospital) to explore the dynamics of Buruli ulcer.

Section-wise, the paper is summarized as follows: The Buruli ulcer model is formulated in section 2 taking into account individuals lost to follow-up. The basic properties, stability and sensitivity analysis of the model as well as some numerical simulations to validate the theoretical results obtained are given in section 3. Basic control of Buruli ulcer using the different mortality thresholds and of bacterial shedding in section 4. The final section 5 concludes the paper and provides some discussions that highlight some relevant perspectives.

In this section, we proceed with the formulation of a mathematical model for the environmental transmission of Buruli ulcer within a human population.

We propose a deterministic compartmental model on the environmental transmission of Buruli ulcer. The population under consideration is grouped into disjoint classes or compartments. We divide the human population into four states representing disease status. At any time t, there are the following epidemiological statuses.

1) Susceptible individuals S. This class contains human individuals who are susceptible to Buruli ulcer.

2) Exposed individuals E. This class contains individuals who have come into contact with the contaminated environment but show no symptoms of the disease. They can after some time develop into illness or heal. Once sick, they can either decide to go to the hospital for treatment or to treat themselves at home.

3) Infectious humans I. The class of human infectious humans contains individuals who, after contracting the infection, progressed to the disease. They went to the hospital to be diagnosed and were isolated for treatment.

4) Lost individuals J. This class contains individuals who, after contracting the infection, progressed to the disease. After being diagnosed with the disease, they left the hospital and decided to treat themselves by other means.

5) Recovered individuals R. This class comprises human cases that recover from the infection, because recovery is accompanied with the acquisition of partial immunity so that this class of individuals is partially protected against further infection and they join the class of susceptible individuals after a period

Thus, the total human population alive at time t is

$N\left(t\right)=S\left(t\right)+E\left(t\right)+I\left(t\right)+J\left(t\right)+R\left(t\right).$(1)

Also, there is an environment class (P), which represents the concentration of Mycobacterium ulcerans in the environment. The environmental class represents the reservoir of contamination in the community associated with individuals lost to follow-up who have decided to seek treatment by means other than the hospital.

The model is based on the following assumptions:

We assume each sub-population of humans is submitted to the natural death rate μ. Susceptible individuals are recruited into the population at rate Λ. Transmission of Buruli ulcer disease within the human population occurs through contact between individuals susceptible to Mycobacterium ulcerans and the Mycobacterium ulcerans concentration in the environment.

We assume that the transmission of Buruli ulcer is modeled by the mass action. Thus, human susceptible individuals are infected at rate $\lambda \left(P\right)S\left(t\right)$ where $\lambda \left(P\right)$ is the force of infection given by

$\lambda \left(P\right)=\gamma P\left(t\right),$(2)

where γ is the effective contact rate for Mycobacterium ulcerans in the environment.

A flow diagram characterizing the full model is depicted in Figure 1 . In the diagram, solid lines represent flow between compartments, the dashed lines represent the infected classes contaminating the environment, and new infections coming from the interaction of susceptibles with the environment.

This transmission model is expressed by the following deterministic system of nonlinear ordinary differential equations:

$\{\begin{array}{l}\stackrel{\dot{}}{S}\left(t\right)=\text{Λ}+\rho R\left(t\right)-\gamma S\left(t\right)P\left(t\right)-\mu S\left(t\right),\hfill \\ \stackrel{\dot{}}{E}\left(t\right)=\gamma S\left(t\right)P\left(t\right)-\left(\epsilon +\mu \right)E\left(t\right),\hfill \\ \stackrel{\dot{}}{I}\left(t\right)=\epsilon p_{1}E\left(t\right)-\left(\mu +\alpha \right)I\left(t\right),\hfill \\ \stackrel{\dot{}}{J}\left(t\right)=\epsilon p_{2}E\left(t\right)-\left(\mu +\theta \right)J\left(t\right),\hfill \\ \stackrel{\dot{}}{R}\left(t\right)=\epsilon \left(1-p_1-p_2\right)E\left(t\right)+\alpha I\left(t\right)+\theta J\left(t\right)-\left(\mu +\rho \right)R\left(t\right)\hfill \\ \stackrel{\dot{}}{P}\left(t\right)=\omega J\left(t\right)-{\mu }_{p}P\left(t\right).\hfill \end{array}$(3)

A complete list and description of all variables and parameters in system (3) is summarized in Table 1 and Table 2 , respectively. For biological reasons, all the parameters are assumed nonnegative.

In this section, we present all the theoretical results of the system (3). We study the basic properties which are essential in the proof of stability results and determine the asymptotic behaviour of our model.

The aim of this section is to study how the mortality rate and excretion of bacteria into the environment impacts the dynamics of Buruli ulcer transmission.

Figure 6 presents the time evolution of basic reproduction number as function of the bacteria in the environment mortality and shedding rate respectively. Ones can see in this figure that the values of the basic reproduction number less or greater than one depends strongly on these two parameters of the model. For the values taken for our numerical simulations, we can estimate a mortality threshold value above which the basic reproduction number R₀ is always less than one: this corresponds to a situation of disease free equilibrium. Biological or chemical control in an aquatic ecosystem can be a means of having a bacterial mortality rate in the environment higher than this threshold value. On the other hand, we have a threshold value for the excretion rate: this means that if we educate the population to be screened and accept hospitalization during treatment, we can limit the excretion rate of the bacteria. in the environment and therefore better control the disease.

More informing the population about the disease, the importance of getting tested, and receiving treatment in hospital, along with biological or chemical control measures, are two approaches that will help us better control the spread of Buruli ulcer among the human population. Thus, to opt for a chemical control method to eliminate the bacteria in the environment, it would be ideal to assess the environmental impact of this measure using the approach proposed in [22] .

In the present work, we present a deterministic mathematical model for the transmission of Buruli ulcer whose aim was to highlight the impact of bacteriumpresent in the environment on the spread of the disease within a human population. Model uses nonlinear differential equations and regroups each human according to their epidemiological status. The particularity of this model is that it considers that once diagnosed, the patient is hospitalized until complete recovery. But the socio-economic nature of our study environment leads people to avoid hospitals even if it means treating themselves: it is this class of population considered as patients lost to sight who therefore contribute to the spread of the infection releasing bacterium into the environment. The obtained model is analysed by determining the steady states. We show that the model analysis is carried out in terms of the basic reproduction number R₀, which summarizes the dynamics of the system: If it is less than one, the disease disappears within a human population; when it is greater than one, the disease is endemic. We use sensitivity analysis that highlights how parameters affects the basic reproduction number R₀ and how parameters affects the output of the variables. Numerical simulations were made to validate the mathematical results obtained. As we observe in Figure 4 and Figure 5 , the numerical results are in agreement with the theoretical results.

Our main result is that the control of the spread of Buruli ulcer within a human population depends mostly on the environmental management and sensitisation of the human population. Lack of knowledge of the disease plays an important role in its spread. the human population must know that, faced with this disease, they must go to the hospital and agree to remain hospitalize until complete recovery: awareness campaigns must be multiplied in this direction. Controlling the spread of the epidemic within the human population remains a challenge. At the end of our study, it suffices to maintain a basic reproduction number ${ℛ}_0$ lower than 1. We carry out sensitivity analysis of ${ℛ}_0$ to find out the degree to which the parameters influence the basic reproduction number ${ℛ}_0$. We show a monotonic decrease of ${ℛ}_0$ when ${\mu }_p$ increase. This means that the clearance of the environment will reduce the spread of Buruli ulcer epidemic. According to Figure 6 , we have identified a threshold for the excretion rate below which we can control the progression of the disease, and a threshold for bacterial mortality above which we can also control the spread of Buruli ulcer among the human population. However, the complete eradication of Mycobacterium in the aquatic ecosystem is not feasible in practice. An approach in our future work could involve combining these factors with the control of water bugs in an optimal control model to determine the best strategy for controlling Buruli ulcer in the population.

Sincere thanks to the members of JAMP for their professional performance, and special thanks to managing editor Hellen XU for a rare attitude of high quality.

Lemma 7. Suppose that a function x of class $C^1\left(I,ℝ\right)$, where I is an interval of $ℝ$, verifies

$\stackrel{\dot{}}{x}\left(t\right)\le a\left(t\right)x\left(t\right)+b\left(t\right),$(16)

where a and b are continuous functions from I into $ℝ$, and $x\left(t\right)=x_0$ for $t_0\in I$. Then we have the inequality

$x\left(t\right)\le x\left(t_0\right)\text{exp}\left({\displaystyle {\int }_{t_0}^{t}a\left(s\right)\text{d}s}\right)+{\displaystyle {\int }_{t_0}^{t}b\left(s\right)\text{exp}\left({\displaystyle {\int }_s^{t}a\left(\sigma \right)\text{d}\sigma }\right)\text{d}s}$(17)

Let $X={\left(S,E,I,J,R,P\right)}^{{\rm T}}$ and consider a Lyapunov function,

$\begin{array}{c}V\left(X\right)=\left(S-\bar{S}-\bar{S}\ln \left(\frac{S}{\bar{S}}\right)\right)+a_1\left(E-\bar{E}-\bar{E}\ln \left(\frac{E}{\bar{E}}\right)\right)\\ +a_2\left(I-\bar{I}-\bar{I}\ln \left(\frac{I}{\bar{I}}\right)\right)+a_3\left(J-\bar{J}-\bar{J}\ln \left(\frac{J}{\bar{J}}\right)\right)\\ +a_4\left(R-\bar{R}-\bar{R}\ln \left(\frac{R}{\bar{R}}\right)\right)+a_5\left(P-\bar{P}-\bar{P}\ln \left(\frac{P}{\bar{P}}\right)\right).\end{array}$(18)

A direct computation leads to

$\begin{array}{c}\stackrel{\dot{}}{V}=\left(1-\frac{\bar{S}}{S}\right)\stackrel{\dot{}}{S}+a_1\left(1-\frac{\bar{E}}{E}\right)\stackrel{\dot{}}{E}+a_2\left(1-\frac{\bar{I}}{I}\right)\stackrel{\dot{}}{I}+a_3\left(1-\frac{\bar{J}}{J}\right)\stackrel{\dot{}}{J}\\ +a_4\left(1-\frac{\bar{R}}{R}\right)\stackrel{\dot{}}{R}+a_5\left(1-\frac{\bar{P}}{P}\right)\stackrel{\dot{}}{P}\\ =\left(1-\frac{\bar{S}}{S}\right)\left[\text{Λ}+\rho R-\gamma PS-\mu S\right]+a_1\left(1-\frac{\bar{E}}{E}\right)\left[\gamma PS-\left(\mu +\epsilon \right)E\right]\\ +a_2\left(1-\frac{\bar{I}}{I}\right)\left[\epsilon p_{1}E-\left(\mu +\alpha \right)I\right]+a_3\left(1-\frac{\bar{J}}{J}\right)\left[\epsilon p_{2}E-\left(\mu +\theta \right)J\right]\\ +a_4\left(1-\frac{\bar{R}}{R}\right)\left[\epsilon \left(1-p_1-p_2\right)E+\alpha I+\theta J-\left(\mu +\rho \right)R\right]\\ +a_5\left(1-\frac{\bar{P}}{P}\right)\left[wJ-{\mu }_{p}P\right].\end{array}$

At the endemic equilibrium point, we have the following relations:

$\{\begin{array}{l}\Lambda =-\rho \bar{R}+\gamma \bar{S}\bar{P}+\mu \bar{S},\hfill \\ \gamma \bar{S}\bar{P}=\left(\epsilon +\mu \right)\bar{E},\hfill \\ \epsilon p_1\bar{E}=\left(\mu +\alpha \right)\bar{I},\hfill \\ \epsilon p_2\bar{E}=\left(\mu +\theta \right)\bar{J},\hfill \\ \epsilon \left(1-p_1-p_2\right)\bar{E}+\alpha \bar{I}+\theta \bar{J}=\left(\mu +\rho \right)\bar{R},\hfill \\ w\bar{J}={\mu }_p\bar{P}.\hfill \end{array}$(19)

Using relations (19), we have

$\begin{array}{c}\stackrel{\dot{}}{V}=\left(1-\frac{\bar{S}}{S}\right)\left[\mu \bar{S}\left(1-\frac{S}{\bar{S}}\right)+\gamma \bar{P}\bar{S}\left(1-\frac{PS}{\bar{P}\bar{S}}\right)+\rho \bar{R}\left(\frac{R}{\bar{R}}-1\right)\right]\\ +a_1\left(1-\frac{\bar{E}}{E}\right)\left[\gamma \bar{P}\bar{S}\left(\frac{PS}{\bar{P}\bar{S}}-\frac{E}{\bar{E}}\right)\right]\\ +a_2\left(1-\frac{\bar{I}}{I}\right)\left[\epsilon p_1\bar{E}\left(\frac{E}{\bar{E}}-\frac{I}{\bar{I}}\right)\right]+a_3\left(1-\frac{\bar{J}}{J}\right)\left[\epsilon p_2\bar{E}\left(\frac{E}{\bar{E}}-\frac{J}{\bar{J}}\right)\right]\\ +a_4\left(1-\frac{\bar{R}}{R}\right)\left[\epsilon \left(1-p_1-p_2\right)\bar{E}\left(\frac{E}{\bar{E}}-\frac{R}{\bar{R}}\right)+\alpha \bar{I}\left(\frac{I}{\bar{I}}-\frac{R}{\bar{R}}\right)+\theta \bar{J}\left(\frac{J}{\bar{J}}-\frac{R}{\bar{R}}\right)\right]\\ +a_5\left(1-\frac{\bar{P}}{P}\right)\left[w\bar{J}\left(\frac{J}{\bar{J}}-\frac{P}{\bar{P}}\right)\right].\end{array}$

Let

$x=\frac{S}{\bar{S}},y=\frac{E}{\bar{E}},z=\frac{I}{\bar{I}},t=\frac{J}{\bar{J}},u=\frac{R}{\bar{R}},v=\frac{P}{\bar{P}},$

$\begin{array}{c}\stackrel{\dot{}}{V}=\left(1-\frac{1}{x}\right)\left[\mu \bar{S}\left(1-x\right)+\gamma \bar{P}\bar{S}\left(1-vx\right)+\rho \bar{R}\left(u-1\right)\right]\\ +a_1\left(1-\frac{1}{y}\right)\left[\gamma \bar{P}\bar{S}\left(vx-y\right)\right]\\ +a_2\left(1-\frac{1}{z}\right)\left[\epsilon p_1\bar{E}\left(y-z\right)\right]+a_3\left(1-\frac{1}{t}\right)\left[\epsilon p_2\bar{E}\left(y-t\right)\right]\\ +a_4\left(1-\frac{1}{u}\right)\left[\epsilon \left(1-p_1-p_2\right)\bar{E}\left(y-u\right)+\alpha \bar{I}\left(z-u\right)+\theta \bar{J}\left(t-u\right)\right]\\ +a_5\left(1-\frac{1}{v}\right)\left[w\bar{J}\left(t-v\right)\right]\\ =\mu \bar{S}\left(\frac{{\left(1-x\right)}^2}{x}\right)+G\left(x,y,z,t,u,v\right)\end{array}$

where

$\begin{array}{c}G\left(x,y,z,t,u,v\right)=\left(1-\frac{1}{x}\right)\left[b_1\left(1-vx\right)+b_2\left(u-1\right)\right]+a_1\left(1-\frac{1}{y}\right)\left[b_1\left(vx-y\right)\right]\\ +a_2\left(1-\frac{1}{z}\right)\left[b_3\left(y-z\right)\right]+a_3\left(1-\frac{1}{t}\right)\left[b_4\left(y-t\right)\right]\\ +a_4\left(1-\frac{1}{u}\right)\left[b_5\left(y-u\right)+b_6\left(z-u\right)+b_7\left(t-u\right)\right]\\ +a_5\left(1-\frac{1}{v}\right)\left[b_8\left(t-v\right)\right]\\ =\mu \bar{S}\left(\frac{{(1-x)}^2}{x}\right)+G\left(x,y,z,t,u,v\right)\end{array}$

and

$b_1=\gamma \bar{P}\bar{S},b_2=\rho \bar{R},b_3=\epsilon p_1\bar{E},b_4=\epsilon p_2\bar{E}$

$b_5=\epsilon \left(1-p_1-p_2\right)\bar{E},b_6=\alpha \bar{I},b_7=\theta \bar{J},b_8=w\bar{J}$

The development of $G\left(x,y,z,t,u,v\right)$ leads to

$\begin{array}{c}G\left(x,y,z,t,u,v\right)=vx\left(-b_1+a_1{b}_1\right)+y\left(-a_1{b}_1+a_2{b}_2+a_3{b}_4+a_4{b}_5\right)\\ +z\left(-a_2{b}_3+a_4{b}_6\right)+t\left(-a_3{b}_4+a_4{b}_7+a_5{b}_8\right)+v\left(b_1-a_5{b}_8\right)\\ +u\left(-a_4{b}_5-a_4{b}_6-a_4{b}_7+b_2\right)+b_1-b_2+a_1{b}_1+a_2{b}_3\\ +a_4{b}_5+a_4{b}_6+a_4{b}_7+a_5{b}_8+\frac{b_2}{x}\left(1-u\right)-\frac{a_1{b}_{1}vx}{y}-\frac{b_1}{x}\\ -\frac{a_1{b}_{1}vx}{z}-\frac{a_3{b}_{4}y}{t}-\frac{a_4{b}_{5}y}{u}-\frac{a_4{b}_{6}z}{u}-\frac{a_4{b}_{7}t}{u}-\frac{a_5{b}_{8}t}{v}\end{array}$

The positive constants $a_1,a_2,a_3,a_4$ and $a_5$ are chosen such that:

$\{\begin{array}{l}-b_1+a_1{b}_1=0,\hfill \\ -a_1{b}_1+a_2{b}_3+a_3{b}_4+a_4{b}_5=0,\hfill \\ -a_2{b}_3+a_4{b}_6=0,\hfill \\ -a_3{b}_4+a_4{b}_7+a_5{b}_8=0,\hfill \\ b_1-a_5{b}_8=0.\hfill \end{array}$(20)

The resolution of system (20) gives:

$a_1=1,a_2=a_4=0,a_3=\frac{b_1}{b_2},a_5=\frac{b_1}{b_8}$

and $G\left(x,y,z,t,u,v\right)$ becomes

$G\left(x,y,z,t,u,v\right)=b_1\left[4-\frac{vx}{y}-\frac{1}{x}-\frac{y}{x}-\frac{t}{v}\right]-b_2\left[1-u-\frac{1}{x}+\frac{u}{x}\right],$

and

$\stackrel{\dot{}}{V}=-\mu \bar{S}\left(\frac{{(1-x)}^2}{x}\right)+b_1\left[4-\frac{vx}{y}-\frac{1}{x}-\frac{y}{x}-\frac{t}{v}\right]-b_2\left[1-u-\frac{1}{x}+\frac{u}{x}\right],$

where

$b_1=\gamma \bar{P}\bar{S},b_2=\rho \bar{R}.$

Using the arithmetic-geometric means inequality leads to

$b_1\left[4-\frac{vx}{y}-\frac{1}{x}-\frac{y}{x}-\frac{t}{v}\right]\le 0$

and $b_2\left[1-u-\frac{1}{x}+\frac{u}{x}\right]>0$. Consequently $\stackrel{\dot{}}{V}\le 0$. By LaSalle’s invariance principle, one can conclude that the endemic equilibrium is globally asymptotically

stable in Ω. Since Ω is absorbing, this proves the global asymptotically stability in the nonnegative orthant.

This concludes the proof.