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Dengue is a flavivirus, transmitted to human through the bites of infected
Aedes
aegypti
and
A. albopictus
mosquitoes. In this paper, we analyze a new system of ordinary differential equations which incorporates saturated incidence function, vector biting rate and control measures at both the aquatic and adult stages of the vector (mosquito). The stability of the system is analysed for the dengue-free equilibrium via the threshold parameter (reproduction number) which was obtained using the Next generation matrix techniques. Routh Hurwitz criterion along together with Descartes’ rule of signs change established the local asymptotically stability of the model whenever
*R*
_{0}
＜1and unstable otherwise. Furthermore, the sensitivity analysis was carried out and the numerical simulation reveals that increasing the proportion of human antibody and putting into place a control strategy that minimize the vector biting rate are enough to reduce the infection of the disease in the population to its barest minimum.

Over the years, mathematical models and computer simulations have been known to be useful experimental tools which are used in building and examing theories, evaluating quantitative speculations, giving answers to particular questions and determining sensitivities to changes in parameter values. Understanding the epidemiology of emerging and re-emerging infectious diseases in a population produces a healthy environment for living. Mathematical models are used in likening, designing, implementing, evaluating and optimizing several detection, prevention and control plans.

Dengue fever is one of the infectious diseases that have continued to be a subject of major concern to the public health. It is known to be a mosquito-borne viral infection which is endemic in more than a hundred countries in the world [

The dengue disease has been well known clinically for over 2 centuries, but the etiology of the disease remains unknown until year 1944 [

Dengue Hemorrhagic fever being an infectious tropical disease is caused by an infective agent called dengue virus, of the family flaviviridae which has four distinguished serotypes denoted by I, II, III and IV [

Dengue infection causes a range of illness in humans, from clinically in apparent, to severe and fatal hemorrhagic disease [

The use of mathematics in explaining the epidemiology of dengue fever has been extensively studied by many researchers over years. Notable among these studies are [

The formulation of dengue model requires the interaction between two-interacting populations (human-vector). The total human population at continuous-time t denoted by N h ( t ) is subdivided into six compartments namely: susceptible humans ( S h ), exposed humans ( E h ), infectious humans ( I h ), migrated population ( M h ), treatment class ( T h ), recovered humans ( R h ). Hence, the total human population N h ( t ) is given by

N h ( t ) = ( S h ) + ( E h ) + ( I h ) + ( M h ) + ( T h ) + ( R h ) (1)

Similarly, the total vector population at continuous-time t denoted by N v ( t ) is subdivided into four compartments namely: aquatic class ( A v ), susceptible mosquitoes ( S v ), exposed mosquitoes ( E v ), infectious mosquitoes ( I v ). Hence, the total vector population N v ( t ) is given by

N v ( t ) = ( A v ) + ( S v ) + ( E v ) + ( I v ) (2)

The dynamics of the dengue considered here is formulated and studied under the following assumption:

1) the model assumes a homogeneous mixing of the human and vector (mosquito) populations, so that each mosquito bite has equal chance of transmitting the virus to susceptible in the population (or acquiring infection from an infected human);

2) considering saturated incidence rate (Non-linear incidence) which incorporate the production of antibodies in response to parasites causing Dengue in both human and vector population ( υ h , υ v ) respectively.

3) the model consider the vector-aquatic class so as to investigate on the effect of the control strategies such as Larvicides at the aquatic stage;

4) that the infectious mosquitoes remain infectious until death;

5) there is loss of immunity for the recovered human population;

6) incorporating the controlling rate parameters which will monitor the effects of control strategies at the aquatic stage ( A v ) and adult stages ( S v , E v , I v ).

In summary, following the assumptions above the transmission dynamics of dengue in a population is given by the following ten compartmental system of non-linear differential equation below:

S ˙ h ( t ) = π h − b β h v S h ( t ) I v ( t ) 1 + υ h I v ( t ) − μ h S h ( t ) + ω R h ( t ) E ˙ h ( t ) = b β h v S h ( t ) I v ( t ) 1 + υ h I v ( t ) + μ 1 M h ( t ) − ( μ h + σ h ) E h ( t ) I ˙ h ( t ) = σ h E h ( t ) + μ 2 M h ( t ) − ( μ h + τ h + δ h ) I h ( t ) M ˙ h ( t ) = π m h − ( μ 1 + μ 2 + μ h ) M h ( t ) T ˙ h ( t ) = τ h I h ( t ) − ( μ h + γ 1 ) T h ( t ) R ˙ h ( t ) = γ 1 T h ( t ) − μ h R h ( t ) − ω R h ( t ) }

A ˙ v ( t ) = π v − ( γ m + μ v + C a ) A v ( t ) S ˙ v ( t ) = γ m A v ( t ) − b β v h S v ( t ) I h ( t ) 1 + υ v I h ( t ) − ( μ v + C m ) S v ( t ) E ˙ v ( t ) = b β v h S v ( t ) I h ( t ) 1 + υ v I h ( t ) − ( θ c + σ v + μ v + C m ) E v ( t ) I ˙ v ( t ) = ( θ c + σ v ) E v ( t ) − ( δ v + μ v + C m ) I v ( t ) } (3)

where a dot is representing differentiation with respect to time.

It is important to explore the basic dynamical feature of the model. For the model (3) formulated above to be epidemiologically meaningful, it is very important to prove that all the states variables non-negative for all time (t). In other words, the solution of the model (3) with positive initial values of data will remain positive at all time t ≥ 0 .

Since model (3) describe interaction between human and vector population, it is important to state that all the parameters and variables involved are non-negative with respect to time. The dengue model (3) will be consider in the biologically-feasible region ℑ = ℑ h × ℑ v ⊂ ℜ + 6 × ℜ + 4 with

ℑ h = { S h , E h , I h , M h , T h , R h ∈ ℜ + 6 : N h ≤ π h μ h } (4)

and