Stability Analysis of Anopheles Mosquito …acadpubl.eu › jsi › 2017-116-23-24 › articles ›...

Stability Analysis of AnophelesMosquito Mathematical Model with the

Unvarying Controller

Naresh Kumar Jothi1 and Suresh Rasappan 2

Department of MathematicsVel Tech University

400 Feet Outer Ring RoadAvadi-Tamilnadu

[email protected]@gmail.com

October 27, 2017

Abstract

In this paper, dynamics of Anopheles mosquito life cyclebreak ups is derived. The equilibrium points of the Anophe-les mosquito life cycle model were found out the bounded-ness of the system were deliberated. The global stability ofthe interior equilibrium points is discussed. The stochasticstability properties of the model are investigated.

AMS Subject Classification: 92B05, 35A24, 93C15,34H15, 34H05

Key Words:Anopheles mosquito, Malaria, Lyapunovfunction, stochastic stability

1 Introduction

The Mosquito is one of the species that give nuisance to the publichealth of the world. It is very powerful to understand the lifecycle

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International Journal of Pure and Applied MathematicsVolume 116 No. 23 2017, 431-444ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

431

of the dangerous mosquito species in which can be controlled tomanage the public health. All mosquito pest will go through com-plete metamorphoses, from egg to larva, from larva to pupa, frompupa to adult. The cycle begins when a female mosquito obtainsa blood meal from either a human-being or from another mammalto supply the required nutrients to produce approximately aroundtwo hundred and fifty eggs at a time [1]. She then seeks an aquaticlocation usually on the surface of stagnant water, or in a waterfilled in depression [2], or on the edge of a container, where rain-water was collected for the female mosquito to lay eggs [3]. Twodays, after the eggs will get hatch into larvae [4]. The larvae liveand feed on microorganisms in the water for seven to fourteen daysand develop into pupae [5], which will not feed for more than four-teen days. The mosquito then emerges from the pupa shell as afully developed adult after four days. The moment the body of themosquito endings moulting, it is hypersensitive to carbon dioxideexhaled from mammals, it has poor eyesight and very sensitive tomammal sweat scent ever from a half mile distance [6], This enablesit to locate a mammal to seek out for a blood meal by a suck onthe animal or human skin. As the mosquito punctures the skin itinjects its saliva into the flesh. If the mosquito is infected with anydisease [7]-[11]. It is transferred into the prey through the saliva.

There are about 2500 species of mosquitoes on the planet, ofwhich 300 are well-known disease carriers. Different species carrydifferent diseases that are local to the area where they live. Thedisease Malaria is transmitted by the female Anopheles mosquitowhich feeds on human blood ([12]-[15]). It is observed that as longas an environment is left uncared for, it will become a breedingground providing many mosquito hatcheries. When an environ-ment is occupied with millions of mosquitoes [16], no preventivemeasures can cure or manage the mosquito pest [17]. The dynam-ics of Anopheles mosquito life cycle breaks-up by using backstep-ping control which was studied ([18]- [20]) and it has been recom-mended that in addition to managing the environment and prevent-ing it from becoming a breeding zone, a permanent control measureshould be employed. This paper investigates the complex effects ofAnopheles mosquito model. The system of Anopheles mosquitowith random noise has not been investigated so far. The presentpaper is a contribution in this unexplored area. Much work has

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International Journal of Pure and Applied Mathematics Special Issue

432

Figure 1: Flow Diagram of Anopheles mosquito life cycle

been done to control this species, but no work has been done so farwith stochastic perturbations.

2 The Mathematical Model

For modelling Anopheles mosquito life cycle, the following assump-tions are made.1. The total population of Anopheles mosquito life cycle consistsof four forms, such as adult, egg, larva and pupa.2. In every stage, the natural death rate is considered uniformly.3. Let bN be the existing population, where b is natural birth rateat adult stage.4. k is the controller in egg stage at the rate α.5. s is controller in larva stage at the rate γ.Figure 1 depicts the flow diagram of Anopheles mosquito life cycle.

The mathematical model of Anopheles mosquito life cycle is

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given belowdx1dt

= bN + ρx4 − (η + µ)x1dx2dt

= ηx1 − (σ + µ)x2dx3dt

= σx2 − (λ+ µ+ β)x3dx4dt

= λx3 − (µ+ ρ)x4

(1)

where x1 is the number of adult mosquito at time t, x2 is thenumber of eggs at time t,x3 is the number of larva at time t, x4 isthe number of pupa at time t, b is the natural birth rate at adultstage, η is the rate of adult mosquito ovisposit, ρ is the rate of pupapush to adult mosquito, µ is the normal death at all the stages, βis the death of larva eat-ups to larva and N is the total population.

3 Analysis of Deterministic Model

In this section, the boundedness, equilibrium, local and global sta-bilities are analyzed. We take up the solution of the system (1)Assume that, x1(0) > 0, x2(0) > 0, x3(0) > 0 and x4(0) > 0 arepositive initial conditions. Let us consider the system (x1) withthese positive initial conditions for local existence, positiveness andboundedness of the solutions. As the right-hand sides of (1) aresmooth functions of (x1, x2, x3, x4) and the parameters, the localexistence and uniqueness properties hold within the positive quad-rant.

The state space for the system (1) is positive quadrant {x1, x2, x3, x4 :x1 > 0, x2 > 0, x3 > 0, x4 > 0} which is an invariant set.

Thus x1(t) > 0, x2(t) > 0, x3(t) > 0 and x4(t) > 0 for all largevalues of t as x1(0) > 0, x2(0) > 0, x3(0) > 0 and x4(0) > 0

Now the following theorem establishes the uniform boundednessof the system (1).Theorem 1: The system (1) is dissipative.

Proof: Let x1(0) > 0, x2(0) > 0, x3(0) > 0 and x4(0) > 0 bethe solution of the system with positive initial conditions.Now we define the function.M(t) = x1(t) + x2(t) + x3(t) + x4(t)Therefore, the time derivative gives

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M(t) = x1(t) + x2(t) + x3(t) + x4(t)

= bN − µx1 − µx2 − µx3 − µx4 − βx3(2)

which impliesdMdt

= bN − ψ(x1 + x2 + x3 + x4)− βx3where ψ = min(1, µ)which gives

dMdt

+ ψM = bN − βx3dMdt

+ ψM ≤ bN + β2 = L(say)dMdt

+ ψM = L

(3)

Now applying the theory of differential inequality (Brikoffand Rota,1982), We obtain

0 ≤M(x1, x2, x3, x4) ≤ Lψ

+ w(x1(0),x2(0),x3(0),x4(0))eψt

and for t→∞, we get0 ≤M(x1, x2, x3, x4) ≤ L

ψ

Thus, all the solutions of the system (1) enter in to the region B,

B = {(x1, x2, x3, x4) ∈ R4+, 0 ≤M ≤ L

ψ+ ∈} for any ∈> 0

which completes the theorem.

4 Equilibria and Local stability Analy-

sis

Let us discuss the boundary and interior equilibrium point. Towardthis end, equating right-hand side of (1) to zero, the following fourequilibrium points are obtained.

(a) The Boundary equilibrium state EP0 = (0, 0, 0, 0)(b)The Boundary equilibrium state EP1 = ( bN

η+µ, 0, 0, 0)

(c)The Boundary equilibrium state EP2 = ( bNη+µ

, bNη(η+µ)(µ+σ)

, 0, 0)

(d) The Boundary equilibrium stateEP3 = (0, bNη

(η+µ)(µ+σ), bNησ(η+µ)(µ+σ)(β+λ+µ)

, 0)

(e) The Boundary equilibrium state E∗P = (x∗1, x

∗2, x

∗3, x

∗4)

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wherex∗1 = (ρ+µ)(µ+σ)(β+λ+µ)

λησα, x∗2 = (ρ+µ)(β+λ+µ)

λσα, x∗3 = (ρ+µ)

λα, x∗4 = α

In this,α = bNλση

(η+µ)(σ+µ)(ρ+µ)(β+λ+µ)−σληρ

5 Global Stability Analysis

Theorem :2 The interior equilibrium point E∗p is globally asymp-

totically stable if

x2λ+ (x1 − x∗1)x1 = (ρ+ µ)x1

x3σ + αs+ (x2 − x∗2)x2 = (β + λ+ µ)x2

x4η + γk + (x3 − x∗3)x3 = (λ+ µ)x3

x1ρ+ bN + (x4 − x∗4)x4 = (η + µ)x4

(4)

which implies

λ =(ρ+µ)x4−(x4−x∗4)x4

x3; σ =

(β+λ+µ)x3−(x3−x∗3)x3x2

; η =(ρ+µ)x2−(x2−x∗2)x2

x1;

ρ =(η+µ)x1+bN+(x1−x∗1)x1

x4;

Proof: Define the Lyapunov function

V (x1, x2, x3, x4) = [(x1 − x∗1)− x∗1In(x1x∗1

)] + [(x2 − x∗2)− x∗2In(x2x∗2

)]

+[(x3 − x∗3)− x∗3In(x3x∗3

)] + [(x4 − x∗4)− x∗4In(x4x∗4

)]

(5)where i = 1, 2, 3, 4, 5 are positive constants to be chosen later It isobserved V is a positive definite function in the region except at E∗

p

where it is zero. Solving the rate of change of V along the solutionsof the system, we get

V (x1, x2, x3, x4) = (x1 − x∗1)[ bN+ρx4x1

+ (η + µ)]

+(x2 − x∗2)[ηx1x2 − (ρ+ µ)]

+[(x3 − x∗3)[σx2x3 − (β + λ+ µ)]

+[(x4 − x∗4)[λx3x4 − (ρ+ µ)]

(6)

V = −(x1 − x∗1)2 − (x2 − x∗2)2 − (x3 − x∗3)2 − (x4 − x∗4)2which implies V is negative. Hence the interior equilibrium pointis globally asymptotically stable.

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6 Stochastic stability analysis of the pos-

itive Equilibrium

Stochastic perturbations were introduced in some of the main pa-rameters involved in the model.In this paper, the stochastic perturbations of the variablesx1, x2, x3andx4are allowed and their values are allowed around the posi-tive equilibriumE∗

p . In this case when it is feasible and locallyasymptotically stable. Local stability of E∗

p is implied by the ex-istence condition ofE∗

p .In Model (1), the stochastic perturbationsof the variables around their value atE∗

p are of white noise type,which is proportional to the distances of x1, x2, x3 andx4 from val-ues x∗1, x

∗2, x

∗3 andx∗4

The equation (1) becomes

dx1 = [bN + ρx4 − (η + µ)x1]dt+ σ1(x1 − x∗1)dwt1dx2 = [ηx1 − (σ + µ)x2]dt+ σ2(x2 − x∗2)dwt2dx3 = [σx2 − (λ+ µ+ β)x3 + αs]dt+ σ3(x3 − x∗3)dwt3dx4 = [λx3 − (µ+ ρ)x4]dt+ σ4(x4 − x∗4)dwt4

(7)

where σi, i=1,2.3,4 and wti , i=1,2.3,4 are independent from eachother standard Wiener process.The dynamical behavior of model (1) is robust on such a kind ofstochasticity by investigating the asymptotic stability behavior ofthe equilibrium E∗

p

. This analysis is mainly to represent the dynamics of the systemaround the interior equilibrium point E∗

p .For this purpose, we lin-earize the model using the following perturbation method, that isthe stochastic differential system of (1) can be centered at its pos-itive equilibrium E∗

pby the change of variablesu1 = (x1 − x∗1); u2 = (x2 − x∗2)u3 = (x3 − x∗3); u4 = (x4 − x∗4)

dx1 = [ρu4 − (η + µ)u1]dt+ σ1u1dw1(t)

dx2 = [ηu1 − (σ + µ)u2]dt+ σ2u2dw2(t)

dx3 = [σu2 − (λ+ µ+ β)u3]dt+ σ3u3dw3(t)

dx4 = [λu3 − (µ+ ρ)u4]dt+ σ4u4dw3(t)

(8)

The linearized SDEs around E∗p take the form

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du(t) = f [u(t)]dt+ g[u(t)]dw(t)where u(t) = [u1, u2, u3, u4]

T and

f(u(t)) =

−(λ+ η) λ 0 00 −(σ + µ) σ 00 0 −(β + λ+ µ) ηρ 0 0 −(ρ+ µ)

u(t)

(9)

g(u(t)) =

σ1u1 0 0 00 σ2u2 0 00 0 σ3u3 00 0 0 σ4u4

(10)

Now the Ito stochastic differential is defined asLV (t) = ∂V (t,u)

∂t+ fTu(t)∂V (t,u)

∂u+ 1

2trace[gT (u(t))∂

2V (t,u)∂u2

g(u(t))]where[ ∂V∂u1, ∂V∂u2, ∂V∂u3, ∂V∂u4

]T

∂2V (t,u)∂u2

= col( ∂2V∂uj∂ui

)

Theorem 3: Suppose the function exists as V ∈ C02(U) satisfyingthe inequalitiesV (t, u) ≤ K2 | u |pLV (t, u) ≤ K3 | u |p, Ki > 0, p > 0Then the trivial solution of ( 1) is globally asymptotically stable.

Theorem 4: The zero solution of (2) is asymptotically meansquare stable when

σ1 >√

2a,σ2, σ4 >

√2r,

σ3, σ5, σ6 >√

2q

Proof: Now consider the Lyapunov function

V (u) =1

2[w1u

21 + w2u

22 + w3u

23 + w4u

24] (11)

where wi are real positive constants are to be chosen in the follow-ing. It is easy to check that inequalities (3) hold with p = 2 . Nowthe Ito process (4) becomes

LV (t, u) = w1[−(η + µ)u1 + ηu2]u1 + w2[−(σ + µ)u2+σu3]u2 + w3[−(β + λ+ µ)u3 + λu4]u3+

w4[ρu1 − (ρ+ µ)u4]u4 + 12trace[gT ∂

2V∂u2

g]

(12)

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here

∂2V

∂u2=

w1 0 0 00 w2 0 00 0 w3 00 0 0 w4

(13)

andgT ∂

2V∂u2

g =

w1σ21u

21 0 0 0

0 w2σ22u

22 0 0

0 0 w3σ23u

23 0

0 0 0 w4σ24u

24

(14)

with12 trace[g

T ∂2V∂u2

g] = 12 [w1σ

21u

21 + w2σ

22u

22

+w3σ23u

23 + w4σ

24u

24]

(15)

LV (t, u) = −w1[η + µ− η u2u1− 1

2σ21]u21

−w2[σ + µ− σ u3u2− 1

2σ22]u22

−w3[β + λ+ µ− λu4u3− 1

2σ23]u23

−w4[ρ+ µ− ρu1u4− 1

2σ24]u24

(16)

which is negative definite.

7 Numerical Simulation And Discussion

In this paper stability analysis of Anopheles mosquito mathemat-ical model with the unvarying controller is analyzed and It is ob-served that the boundary equilibrium point is feasible. If the deathrate of the mosquito population remains a certain threshold value,then the positive equilibrium is feasible. Moreover, all the solutionconverges to the positive equilibrium. It is observed that the deter-ministic model is robust on the stochastic perturbations. It is tobe noted that when x1, x2, x3 and x4increases.For the numerical simulations, the fourth order Runge-Kutta methodis used. The parameter values are taken as a = 0.2; b = 0.9; c =2000; d = 5000; q = 0.9; β = 0.5;σ = 0.6; ρ = 0.7;µ = 0.6; η =0.5;λ = 0.4 and the initial densities are taken as x1 = 31759, x2 =13982, x3 = 78366, x4 = 39877 .From this analysis, it is observed that the large value of N com-pared to the controllers, the system takes a larger time to bring the

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439

0 5 10 15 20 25 30 35 40 45 50

1

2

3

4

5

6

7

8x 10

4

Time

x1, x2, x3, x4

x

1− Adult

x2− Egg

x3− Larva

x4− Pupa

Figure 2: stabilization of stochastic Anopheles mosquito modelwhen the total population N = 100000000

mosquito population under short run. In Figure 2, depicts the sta-bilization of stochastic Anopheles mosquito model when the totalpopulation N = 100000000.

8 Conclusion

In this paper, stability analysis of Anopheles mosquito mathemati-cal model with unvarying controller is analyzed. The boundednessof the model has been found and the equilibrium points of the sys-tem have been identified. Global stability properties of the modelare investigated by using the Lyapunov function. The stochasticperturbations are introduced and suggested the deterministic modelis robust on stochastic perturbations. It is showed that the interiorequilibrium point of the Anopheles mosquito model is global asymp-totically stable by constructing suitable Lyapunov function. More-over, all the solutions converge to the positive equilibrium. Thestochastic perturbation is also introduced by to the system. Usingstochastic differential equations and Ito process, it is showed that

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the zero solution of this stochastic system is asymptotically meansquare stable through the construction of the Lyapunov function.Finally, numerical examples are given and diagrams are presentedwhich support the results.

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