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This paper examines optimal control of transmission dynamics of
*Mycobacterium ulceran* (MU) infection. A nonlinear mathematical model for the problem is proposed and analysed qualitatively using the stability theory of the differential equations, optimal control and computer simulation. The basic reproduction number of the reduced model system is obtained by using the next generation operator method. It is found that by using Ruth Hurwitz criteria, the disease free equilibrium point is locally asymptotically stable and using centre manifold theory, the model shows the transcritical (forward) bifurcation. Optimal control is applied to the model seeking to minimize the transmission dynamics of MU infection on human and water-bugs. Pontryagin’s maximum principle is used to characterize the optimal levels of the controls. The results of optimality are solved numerically using MATLAB software and the results show that optimal combination of two controls (environmental and health education for prevention) and (water and environmental purification) minimizes the MU infection in the population.

Mycobacterium ulceran (MU) is a pathogenic, toxin-producing bacterium that is the causative agent of Buruli ulcer (BU), a necrotizing skin infection in humans [

The BU disease infects the skin and subcutaneous tissues resulting in indolent ulcers, with lesions appearing mainly in the limbs. The ulcers grow slowly and release a toxin which damages the skin and underlying tissue. The toxin produced by the causative organism is named Mycolatone, a class of polyketides derived from manolides. The toxin destroys large areas of the skin after manifesting itself in the form of painless dermal nodules [

The mode of transmission of MU currently is unclear for many scholars. There are some hypotheses that have been proposed in connection to the mode of transmission of MU. One of the hypotheses says that the microbe is transmitted through the aquatic environment, whereas MU could infect humans who have frequent contact with contaminated water through swimming or through body injuries that facilitate the introduction of the microbe into the skin. Another hypothesis suggested that MU can be transmitted through the bite of aquatic bugs [

Some studies have exposed various methods of controlling the MU which cause Buruli ulcer (BU). The studies include Mycobacterium bovis basillus Calmetle-Guerin BCG vaccination as prophylaxis against Mycobacterium ulcerans osteomyelitis in Buruli Ulcer Disease for which it recommends BCG vaccination at birth as a control mechanism [

This section investigates the dynamics of Mycobacterium ulceran in a human population as well as that of the vector population. Environmental factors such as arsenic (As) concentration have an influence on the disease prevalent in the population. To understand the transmission dynamics of MU in a population, a mathematical model is developed and analysed. The model discussed describes the dynamics of the two different populations that interact and cause the spread of the disease.

In formulating the model, the following assumptions are taken into consideration:

1) MU infection can arise to the population when there is interaction between human and water bugs.

2) Person to person transmission is excluded.

3) Seasonal variations in the life cycle of the water-bug are negligible.

4) Human population and water-bug populations are homogeneous.

5) Human population is constant.

6) Whenever humans are within the vicinity of the breeding grounds of the water-bugs, they are randomly bitten by the bugs.

The proposed model subdivides the population of interest into two sub populations; human population and vector population. Human population “

Taking into account the above considerations, we have the following schematic flow diagram for the model without control.

The dynamics of the groups described above and as shown in the model flow chart (

Since human population is constant and water-bugs seasonal variation is neglected, then we can analyse the three classes of infected human, infected water-bugs and water contamination.

Let

Let again

We substitute Equation (3) and Equation (4) into equation systems (2) to get

Let

The reduced model system of Equation (6) will be analysed qualitatively to understand the transmission dynamics of MU infection in a population. Threshold which governs persistence of the MU infection will be determined.

The disease free equilibrium point of the reduced model system (6) is obtained by setting

Thus we have

Since we are dealing with disease free equilibrium then we set

Therefore the Disease Free Equilibrium (DFE) denoted by

The basic reproduction number, denoted by

where,

From the equations system (6), it follows that

By linearization approach, the associate matrix at disease free equilibrium is obtained as

This is equivalent to

The Jacobian matrix of the system (13) at the disease free equilibrium point

The transfer of individuals out of the compartment i is given by

The Jacobian matrix of

This gives

with

Thus

Thus the eigenvalues of

Then the effective reproduction number which is given by the largest eigenvalue for the reduced model system (6) is given by

In determining how best to reduce human mortality and morbidity due to MU infection, the sensitivity indices of the reproduction number

Definition 1. The sensitivity index of a variable “p” that depends differentiable on a parameter “q” is defined as:

Having an explicit formula for

respectively. Other indices

From

Parameter Symbol | Sensitivity Index | |
---|---|---|

1 | +0.500000000 | |

2 | −0.49999999 | |

3 | −0.500000001 | |

4 | +0.50000000 |

they accelerate the transmission of MU in the population as they have positive indices. While the parameters

Specifically, the most sensitive parameter is recruitment/death rate of water-bugs

The stability of disease free equilibrium point

The characteristic equation corresponding to

where;

The three eigenvalues have negative real parts if they satisfy the Routh-Hurwitz Criteria, that is;

If

It was shown that

The endemic equilibrium points (EEP) of the reduced model equation system (6) is given by

where

From the Equation (26) it follows that

We can check from the quadratic (26) for the possibility of existence of multiple equilibria. It is important to note that the coefficient A is always positive and C is positive if

There are precisely two endemic equilibria if

From this result we state the theorem which will be proved by using bifurcation diagram and centre manifold theorem.

Theorem 1. The two endemic equilibrium points

The existence of endemic equilibrium which is locally stable for

From

The local asymptotic stability of endemic equilibrium is analysed by using the centre manifold theory [

The model without control was formulated using a system of ordinary differential equations. The model was qualitatively analysed for the existence and stability of the disease-free equilibrium point

The model was further extended with incorporation of control so as to reduce the transmission dynamics of MU infection.

Now the model Equations (6) is extended to incorporate time-dependent controls to obtain the following system:

In the system (27) two control variables

It is intended to minimize the MU infection on human caused by the interaction between susceptible human and infected vector (water-bugs), as well as minimizing MU infection on vector (water-bugs) caused by water contamination. To investigate the optimal level of effort that would be needed to control the disease, first we formulate the objective functional J which is defined by choosing a quadratic cost on the controls as follows:

where

The choice of quadratic control in the objective function is simply because we need to minimize the MU infection as well as minimize the cost on the control. The goal is to minimize the MU infection in human population and in water-bugs while minimizing the cost of controls

where the control set

The term

where

Theorem 2. There exists an optimal control

with the transversality conditions

To find

We differentiate Equation (21) with respect to

We therefore solve for

By equating system (24) to zero we obtain

From the system (23) then

By standard control arguments involving the bounds on the controls, we conclude similarly as [

According to the prior boundedness of the state system, the adjoint system and the resulting Lipschitz structure of the ODEs the uniqueness of the optimal control for small

There is a restriction on the length of time interval in order to guarantee the uniqueness of the optimality system. This smallness restriction of the length on the time due to the opposite time orientations of the optimality system; the state problem has initial values and the adjoint problem has final values. This restriction is common in control problems [

In order to illustrate the analytical results of the study, numerical simulations of the model equations with control variables (27) are carried out using the set of parameter values below:

In Figures 3-6, we use the following weight factors throughout,

From

Parameters | Values per month | Source |
---|---|---|

0.05 | Estimated | |

a | 100 | [ |

0.03 | [ | |

0.15 | [ | |

0.0014 | Estimated | |

0.0015 | [ | |

0.002 | Estimated |

For the control profile as shown in

For the control profile as shown in

Control profile in

the final time while

In this paper, a deterministic model for the transmission dynamics of MU infection was derived and analysed. The model incorporates the assumption that MU infection arises in the population through the interaction of human population and water-bugs population (susceptible human interacting with infected water-bugs or susceptible water-bugs interacting with infected human). The basic reproduction number

Magreth AngaKimaro,Estomih S.Massawe,Daniel OluwoleMakinde, (2015) Modelling the Optimal Control of Transmission Dynamics of Mycobacterium ulceran Infection. Open Journal of Epidemiology,05,229-243. doi: 10.4236/ojepi.2015.54027