^{1}

^{1}

^{2}

This paper examines the effect of treatment of Dengue fever disease. A non linear mathematical model for the problem is proposed and analysed quantitatively using the stability theory of the differential equations. The results show that the disease-free equilibrium point is locally andglobally asymptotically stable if the reproduction number
(R_{0})
is less than unity. The additive compound matrices approach is used to show that the dengue fever model’s endemic equilibrium point is locally asymptotically stable when trace, determinant and determinant of second additive compound matrix of the Jacobian matrix are all negative. However, treatment will have a control of dengue fever disease. Numerical simulation of the model is implemented to investigate the sensitivity of certain key parameters on the dengue fever disease with treatment.

Dengue is a vector borne disease transmitted to humans by the bite of an infected female Aedes mosquito [

A non linear mathematical model is formulated and analysed showing the effect of treatment of Dengue fever disease. The basic reproduction number and stability of equilibrium points are analysed qualitatively. Sensitivity analysis of parameters and numerical simulations are performed. The total human population at any time t will be denoted by

Thus

where

There are three other state variables, related to the female mosquitoes, indexed by

In formulating the model, the following assumptions are considered:

i) Total human population

ii) The population is homogeneous, which means that every individual of a compartment is homogeneously mixed with the other individuals,

iii) Immigration and emigration are not considered,

iv) Each vector has an equal probability to bite any host,

v) Humans and mosquitoes are assumed to be born susceptible i.e. there is no natural protection,

vi) The coefficient of transmission of the disease is fixed and does not vary seasonally,

vii) For the mosquito there is no resistant phase, due to its short lifetime, ([

Considering the above assumptions, we then have the following

Schematic model flow diagram for dengue fever disease with treatment:

From

where

The model system of Equation (1) will be analysed qualitatively to get a better understanding of the effects of treatment of Dengue fever disease. The basic Reproduction number

For the disease free equilibrium, it is assumed that there is no infection for both populations of human and mosquitoes i.e.

The basic reproduction number, denoted by

The basic reproduction number of the model (1)

where,

Using the linearization method, the associated matrix at DFE is given by

This implies that

With

or

The transfer of individuals out of the compartment

Using the linearization method, the associated matrix at DFE is given by

This gives

Therefore

The eigenvalues of the Equation (3) are given by

This gives

It follows that the Reproductive number which is given by the largest eigenvalue for model system (1) with treatment denoted by

where

If

In order to determine how best human mortality due to dengue fever is reduced, we calculate the sensitivity indices of the reproduction number

Definition 1: The normalized forward sensitivity index of “

“

As we have an explicit formula for

The parameters are ordered from most sensitive to the least.

Interpretation of Sensitivity IndicesFrom

But individually, the most sensitive parameter is the average daily biting (per day)

To determine the local stability of the disease free equilibrium, the variation matrix

where

Therefore the stability of the disease free equilibrium point can be clarified by studying the behaviour of

Parameter symbol | Sensitivity index | |
---|---|---|

1 | 1.000002246 | |

2 | 0.511849685 | |

3 | 0.500000701 | |

4 | 0.50000028 | |

5 | 0.5000001122 | |

6 | 0.021327046 | |

7 | −0.00005268151704 | |

8 | −0.011848351 | |

9 | −0.499947596 | |

10 | −1.021327442 |

when

The other eigenvalues are given as

when

when

and finally

when

Hence under certain conditions the system is stable since all the seven eigenvalues are negative. These imply that at

In this subsection, we analyse the global behaviour of the equilibria for system (1). The following theorem provides the global property of the disease free equilibrium

Theorem 1: If

Proof:

To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function:

Calculating the time derivative of

Then substituting

where

It follows that

or

which is equivalent to

But

or

Substituting (9) into (8) yields

Therefore

Thus,

Since we are dealing with presence of dengue fever disease in human population, we can reduce system (1) to a 3-dimensional system by eliminating

The endemic equilibrium of the system (10) is given by

For the existence and uniqueness of endemic equilibrium

Adding Equations (11)-(13) above, we have

or

But from (13) above

It follows that

or

Consequently

Then

This imply that

and

meaning that

Thus, the endemicity of the disease exists since

In order to analyse the stability of the endemic equilibrium, the additive compound matrices approach is used, using the idea of ([

If

Local stability of the endemic equilibrium point is determined by the variational matrix

The following lemma was stated and proved by [

Lemma 1: Let

Using the above Lemma, we will study the stability of the endemic equilibrium.

Theorem 2: If

Proof:

From the Jacobian matrix

Thus,

Using Mathematica software, we get

Hence trace and determinant of the Jacobian matrix

The second additive compound matrix is obtained from the following lemma.

Lemma 2: Let

Proof:

The sub matrix of

The sub matrix of

The coefficient of

Other entries were done following the same method and to obtain

Thus

Using Mathematica software, we get

Therefore

Thus, from the lemma 1, the endemic equilibrium

Here, we illustrate the analytical results of the study by carrying out numerical simulations of the model system (1) using the set of estimated parameter values given as shown below.

starting values in three cases as shown below [

The equilibrium point of the endemic equilibrium

It is observed from

Figures 3(a)-(d) show the variation of population in different classes, human susceptibles, treated human infective, dengue fever patient for different values of

From

ulation. The proportion of Dengue fever disease infectives decreases in time then reaches equilibrium due to the increase in the number of population getting treatment. Moreover treated infectives increase and then decrease due to infected population moving to other classes, and then also recovery population increases in time as more population are treated. Furthermore infected mosquitoes decrease when the recovery population increases. Mosquito susceptible and aquatics increase with time and reaches its equilibrium point due to its short life span.

From

From

From

From

From

patients get treatment which prolongs their lives.

A nonlinear mathematical model has been analysed to study the effect of treatment on the dengue fever disease. The analysis of the model shows that the disease-free equilibrium is locally asymptotically stable by next generation method, which involves the computation of basic reproduction number

demic equilibrium point, that is locally asymptotically stable when

are all negative, then all eigenvalues of

Numerical results are provided to illustrate the analytical results. Sensitivity analysis shows that the average daily biting (per day)

In numerical simulation it is observed that the increase of average daily biting (per day), tends to increase the number of infectious individual in the population. But the absence of average daily biting (per day), the infectious population is lowered and the disease can be eradicated. Moreover the increase of treatment will result the reduction of infected proportion as infected proportion population will move to other class, on the other hand when treatment is applied majority of infectious will be observed, as treatment will prolong the life of individual, but with no treatment infectious will be reduced because majority will die and will reach at equilibrium point. From this indicate that there is much work to be done to eradicate the disease by driving reproduction number to be less than unity. Thus the best thing to do is spraying pesticides to kill mosquitoes or sterile male mosquitoes as biological control.

A compartmental model for Dengue fever disease was presented, a model based on two populations, humans and mosquitoes with treatment. Simulation shows that on the application of treatment, the number of death is reduced. It has been proved algebraically that, if a constant minimum level of a treatment is applied, it is possible to maintain the basic reproduction number below unity, and the infected humans were smaller.