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We consider a rift valley fever model with treatment in human and livestock populations and trapping in the vector (mosquito) population. The basic reproduction number R
_{0} is established and used to determine whether the disease dies out or is established in the three populations. When R
_{0} ≤ 1, the disease-free equilibrium is shown to be globally asymptotically stable and the disease does not spread and when R
_{0} > 1, a unique endemic equilibrium exists which is globally stable and the disease will spread. The mathematical model is analyzed analytically and numerically to obtain insight of the impact of intervention in reducing the burden of rift valley fever disease’s spread or epidemic and also to determine factors influencing the outcome of the epidemic. Sensitivity analysis for key parameters is also done.

Rift Valley Fever (RVF) is an infectious disease caused by the RVF virus of the genus Phlebovirus and family Bunyaviridae. It is transmitted between animal species, including cattle, sheep, goats, and camels, primarily through the bite of the female mosquito, usually Aedes or Culex [

The rest of the paper is arranged as follows. In Section 2, we formulate the mathematical model and establish the basic properties of the model. In Section 3, we compute the basic reproduction number herein referred to as the effective reproduction number, and determine the local and global stability of the Disease Free equilibrium. In Section 4, we establish the existence and stability of the Endemic Equilibrium. In Section 5, we have sensitivity analysis with its interpretation. Section 6 has numerical simulation and Section 7 is the conclusion.

In this model we divide the three populations into the susceptible,

with initial conditions,

are given by _{1}, β_{2}, β_{3}, and β_{4} are the trans-

mission rates. Adding equations system 1, we have

In this section, we carry out stability analysis of the model (1). The model properties are employed to establish criteria for positivity of solutions and well-possessedness of the system.

In this section a region in which solutions of the model system (4.1) are uniformly bounded in a proper subset

so

where

Hence,

Lemma

The region

Lemma

Let the initial data be

Proof From the first equation of the model system 1

that is

integrating by the equation above gives,

Then

Similarly, it can be shown that the remaining eight equations of system (4.1) are also positive

In this section the model system (4.1) is qualitatively analysed by determining the equilibria, carrying out their corresponding stability analysis and interpreting the results. Let

From the second, fourth and sixth equations of (4), we write

defining

Equation (2) reduces to (3).

This solution

In this section, the threshold parameter that governs the spread of a disease referred to as the effective reproduction number is determined. Mathematically, it is the spectral radius of the next generation matrix [

From the system (6),

substituting

The partial derivatives of

The eigenvalues of

The effective reproduction number

The disease-free equilibrium point is

system (1) is computed by differentiating each equation in the system with respect to the state variables

where

Using Birkhoff and Rota's theorem on the differential inequality (3) we obtain

From the matrix (7) we note that the first, third, fourth, fifth and sixth have diagonal entries. Therefore their corresponding eigenvalues are;

With the help of mathematical software, the following characteristic equation is obtained

and

If

Theorem

The disease-free equilibrium point is locally asymptotically stable if

In the presence s of infection, that is,

where

We let

Adding the last two equations of the system and making some simplifications we obtain

where

The equation,

1) If

2) If

3) Otherwise, there is none.

However it is important to note that A is always positive if

Theorem 5 The rift valley fever basic model has,

1) Precisely one unique endemic equilibrium if

2) Precisely one unique endemic equilibrium if

3) Precisely two endemic equilibrium if

4) None, otherwise.

From (iii) it is observed that backward bifurcation is possible if the discriminant is set

where backward bifurcation occurs for values of

Theorem 5 The endemic equilibrium point,

Sensitivity analysis determines parameters that have a high impact on

The indices are crucial and will help us determine the importance of each individual parameter in transmission dynamics and prevalence of the Rift Valley Fever Virus.

Definition 1 The normalized forward sensitivity index of a variable, u, that depends differentiably on index

on a parameter, p is defined as;

The analytical expression for the sensitivity of

volved in

From

We carry out numerical simulations for mathematical model of rift valley fever for the set of parameters from literature as shown in

We have the following simulation results (Figures 1-6).

Parameter symbol | Value | Sensitivity Index |
---|---|---|

π_{m} | 100,000 | −0.0000023 |

π_{l} | 100,000 | −0.000139643 |

μ_{m} | 0.8 | 0.108334 |

γ_{m} | 0.9 | 0.131592 |

γ_{h} | 0.5 | −0.186422 |

β_{2} | 2.9 | 0.133382 |

γ_{l} | 0.5 | −0.215213 |

π_{h} | 100,000 | 0.322104 |

μ_{l} | 0.2 | −1.191842 |

μ_{h} | 0.2 | −1.191842 |

δ_{l} | 0.5 | −1.191842 |

δ_{h} | 0.5 | −1.191842 |

β_{l} | 0.531 | 2.48506 |

In this section parameters,

The Rift Valley Model formulated in this study is well posed and exists in a feasible region where disease free and endemic equilibrium points are obtained and their stability investigated. The model has two interventions; treatment for human and livestock and trapping for mosquitoes. We have shown that disease free equilibrium exist and is locally asymptotically stable whenever its associated effective reproduction number

In the absence of treatment of human or livestock and trapping for mosquitoes:

With human or livestock and trapping for mosquitoes

In this paper, the rift valley fever model with interventions was formulated and analysed. Using the theory of differential equations, the invariant set in which the solutions of the model are biologically meaningful was derived. Boundedness of solutions was also proved. Analysis of the model showed that there exist two possible solutions, namely the disease-free point and the endemic equilibrium point. Further analysis showed that the disease-free point is locally stable implying that small perturbations and fluctuations on the disease state will always result in the clearance disease if

Jonnes Lugoye,Josephine Wairimu,C. B. Alphonce,Marilyn Ronoh, (2016) Modeling Rift Valley Fever with Treatment and Trapping Control Strategies. Applied Mathematics,07,556-568. doi: 10.4236/am.2016.76051