1. IJAMSS - Modeling and Analysis of a Vector-Host Epidemic Model - Sneha Porwal - OPaid

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MODELING AND ANALYSIS OF A VECTOR-HOST EPIDEMIC MODEL WITH

SATURATED INCIDENCE RATE UNDER TREATMENT

SNEHA PORWALsup1 S JAINsup2 R KHANDELWAL3 amp G UJJAINKAR4

13School of Studies in Mathematics Vikram University Ujjai Madhya Pradesh India

sup2Government College Kalapipal District Shajapur Madhya Pradesh India

4Govt P G College Dhar Madhya Pradesh India

ABSTRACT

Global stability of an epidemic model for vector-borne disease was studied by Yang et al [7] A reinvestigation of

the model with a saturated incidence rate and a treatment function proportionate to infectious population I is presented to

understand the effect of the capacity for treatment An equivalent system is obtained which has two equilibriums a

disease-free equilibrium and an endemic equilibrium The stability of these two equilibriums can be controlled by the basic

reproduction number 0real The global stability of the disease-free equilibrium state is established by Lyapunov method and

a geometric approach is used for the global stability of the endemic equilibrium state The model has a globally

asymptotically stable disease-free solution whenever the basic reproduction number 0real is less than or equal unity and has

a unique positive globally asymptotically stable endemic equilibrium whenever 0real exceeds unity Numerical examples

are given for the model with different values of the parameters Graphical presentations are also provided The details are

supplemented by numerical results given in annexure

KEYWORDS Epidemic Model Vector-Borne Disease Saturated Incidence Equilibrium Point Stability Reproduction

Number Treatment Function

AMS Subject Classification 92D25 92D30

1 INTRODUCTION

The main purpose of this paper is to study the dynamics of Vector-borne disease and understand the effect of the

capacity for treatment Vector-borne diseases are infectious diseases caused by viruses bacteria protozoa or rickettsia

which are primarily transmitted by disease transmitting biological agents (anthropoids) called vectors who carry the

disease without getting it themselves Globally malaria is the most prevalent vector-borne disease whose vectors are the

mosquitoes The mosquitoes are vectors of a number of infectious diseases most prominent among which are dengue (the

second most important vector-borne disease) yellow fever St Louis Encephalitis Japanese Encephalitic and West Nile

Fever caused by the West Nile Virus The literature dealing with the mathematical theory and dynamics of vector-borne

diseases are quite extensive Many mathematical models concerning the emergence and reemergence of the vector-host

infectious disease have been proposed and analyzed in the literature [9 10] The mathematical models of epidemiology for

the West Nile virus and for dengue have been investigated in [1 3] and [10 11]

Mathematical modeling became considerable important tool in the study of epidemiology because it helped us

International Journal of Applied Mathematics amp

Statistical Sciences (IJAMSS)

ISSN(P) 2319-3972 ISSN(E) 2319-3980

Vol 4 Issue 6 Oct - Nov 2015 1-16

copy IASET



2

Sneha Porwal S Jain R Khandelwal amp G Ujjainkar

Impact Factor (JCC) 20346 NAAS Rating 319

to understand the observed epidemiological patterns disease control and provide understanding of the underlying

mechanisms which influence the spread of disease and may suggest control strategies The model formulation and its

simulation with parameter estimation allow us to test for sensitivity and comparison of conjunctures Treatment plays an

important role to control or decrease the spread of diseases Kar et al [1617] and Ujjainkar et al [4] proposed an epidemicmodel with treatment function Wei et al [6] investigated an epidemic model of a vector-borne disease with direct

transmission and the vector-mediated transmission The global stability of an epidemic model for vector-borne disease was

studied by Yang et al [7] Here we have reanalyzed the model [7] with saturated incidence and a treatment function

proportionate to infectious population I The only mode of transmission in this model is through the vector

The paper is organized as follows In Section 2 a vector-host epidemic model with vector transmissions is

presented where the dynamics of the hosts and vectors are described by SIR and SI model respectively Equilibrium points

and reproduction number are obtained in Section 3 The analysis of stability of the equilibrium of the model is investigated

in Section 4 and Section 5 Numerical analysis and conclusion are given in Section 6 and Section 7

2 THE MATHEMATICAL MODEL

In this section a vector-host epidemic model with vector transmissions is presented and investigated where the

dynamics of the human hosts and vectors are described by SIR and SI model respectively The total host population

1( ) N t is partitioned into three distinct epidemiological subclasses which are susceptible infectious and recovered with

sizes denoted by ( )S t ( ) I t and ( ) R t respectively and the total vector population 2( ) N t is divided into susceptible

and infectious with the sizes denoted by ( ) M t and ( )V t respectively

The dynamics of this infectious disease in the host and vector populations are described by the following system

of nonlinear diff erential equations

1 1

1

1

1

1

2 2 2

2 2

1

1

(1 )

(1 )

dS b S

dt

dI I I rI

dt

dR I R rI

dt

d

SV

V

S

M b MI M dt

dV MI

V

V

V dt

micro α

γ micro α

γ micro

λ micro

λ

λ

λ micro

= minus minus

= minus minus minus

= minus +

= minus minus

= minus

+

+

(21)

With the initial Conditions 0(0)S S = 0

(0) I I = 0(0) R R= 0

(0) M M = and 0(0)V V =

The parameters in the model stand for

1b = Recruitment rate constant in the host population

2b = Recruitment rate constant in the vector population



Modeling and Analysis of a Vector-Host Epidemic Model with Saturated Incidence Rate Under Treatment 3


1 micro = Death rate constant in the host population

2 micro = Death rate constant in the vector population

γ = Recovery rate constant in the host population

1λ = Transmission rate from infected vector population to susceptible host population

2λ = Transmission rate from infected host population to susceptible vector population

1α =Constant parameter

r = Treatment rate constant in the host population

It is assumed that all parameters are positive The total dynamics of host population can be determined from the

differential equation 1 1 1 1 dN dt b N micro = minus

which is derived by adding first three equations of system (21) The total

dynamics of vector population can be determined from the differential equation 2 2 2 2 dN dt b N micro = minus

which is

derived by adding last two equations of system (21) It is easily seen that both for the host population and for the vector

population the corresponding total population sizes are asymptotically constant 1 1 1lim t

N b micro rarrinfin

= and

2 2 2lim t

N b micro rarrinfin

= This implies that in our model we can assume without loss of generality that 1 1 1 N b micro =

2 2 2 N b micro = for all t ge 0 provided that 1 1(0) (0) (0) S I R b micro + + = 2 2(0) (0) M V b micro + =

Therefore we attact system (21) by studying the subsystem given by

1 1

1

1

1

2

2

1

2

1

2

(1 )

(1

)

SV

V

SV

V

dS b S

dt

dI I I rI

dt

bdV V I V

dt

micro α

γ micro α

micro

λ

λ

λ micro

= minus minus

= minus minus minus

minus= minus

+

+

(22)

From biological considerations we study the system (22) in the closed set

Γ = (S I V ) isin3

R+ 1 10 S I b micro le + le 2 20 0 0V b S I micro le le ge ge

Where3 R+ denotes the nonnegative cone of

3 R including its lower dimensional faces Obviously Γ is

positively invariant set of (22)We denote by partΓ and0Γ the boundary and the interior of Γ in

3 R respectively

3 MATHEMATICAL ANALYSIS OF THE MODEL

Equilibrium points of model (22) can be obtained by equating right hand side to zero The system (22) has two



4



equilibrium states the disease-free equilibrium 0 E = ( 1

1

b

micro 0 0) isin partΓ and a unique endemic equilibrium

E lowast

= ( ) S I V lowast lowast lowast 0isin Γ with

( ) ( )1 1

1

1

I r V S

V

γ micro α

λ

lowast

lowast + + +

= (31)

( )

2

1 2 1 2 1 2 1

2 1 1 2 1 2 1 2

( )

( )

b b r I

r b b

λ λ micro micro γ micro

λ γ micro λ micro micro α

lowast minus + +=

+ + + +

(32)

( )2 2

2 2 2

b I

V I

λ

micro micro λ

lowastlowast

lowast=

+

(33)

The dynamics of the disease are described by the basic reproduction number 0real The threshold quantity 0

real is

called the reproduction number which is defined as the average number of secondary infections produced by an infected

individual in a completely susceptible population The basic reproduction number of model (22) is given by the expression

1 2 1 20 2

1 2 1

( )

b b

r

λ λ

micro micro γ micro real =

+ + (34)

For 01real le the only equilibrium is disease-free equilibrium 0

E in partΓ and for 01real gt there is a unique endemic

equilibrium E lowast in 0Γ

4 STABILITY OF THE DISEASE-FREE EQUILIBRIUM

In this section we discuss the stability of the disease-free equilibrium 0 E

Theorem 41 The disease-free equilibrium 0 E

of (22) is locally asymptotically stable in Γ if 0

1real lt it is

unstable if 01real gt

Proof To discuss the stability of the system (22) the variational matrix is

( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( )1 1

0

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

= minus + + + +

minus minus +

V S

V V

V S J r

V V

bV I

At the equilibrium point 0 E = ( 1

1

b

micro 0 0) the variational matrix becomes





1 11

1

1 10 1

1

2 22

2

0

( ) 0 ( )

0

λ micro

micro

λ γ micro

micro

λ micro

micro

minus minus

= minus + +

minus

b

b J E r

b

Its characteristic equation is

( ) 2 1 2 1 21 1 2 2 1

1 2

[ ( ) ( ) ] 0λ λ

λ micro λ γ micro micro λ micro γ micro micro micro

+ + + + + + + + minus =bb

r r

( ) ( )21 1 2 2 1 0

[ ( ) ( ) 1 ] 0λ micro λ γ micro micro λ micro γ micro rArr + + + + + + + + minus real =r r

(41)

By Descartes rule of sign all roots of equation (41) are negative if 01real lt Thus if 0

1real lt then the disease-free

equilibrium 0 E is locally asymptotically stable Otherwise if 0

1real gt then it is unstable

Theorem 42 If 01real le then the disease - free equilibrium 0

E is globally asymptotically stable in Γ

Proof To establish the global stability of the disease-free equilibrium we construct the following Lyapunov

function

1 2 1 1 L I bV micro micro λ = +

Clearly 0 L ge along the solutions of the system (22) and is zero if and only if both I and V are zero

Calculating the time derivative of L along the solutions of system (22) we obtain

( )

( )

( ) ( )

1 2 1 1

2

1 2 1 1 2 21

21

1 1 2 1 21 2 1 2 1 2 1 1 1 21

1 2

1

1

1

2 0 1 2 11

(1

1

0

)

= +

minus= minus + minus+ +

le minus minus minus minus+ +

= minus minus real + + +

+

le

L I b V

bV I b I V r

b bb I bVI b V r

SV

V

b

V

I V r

λ

λ

micro micro λ

micro micro λ λ micro γ micro micro α

λ λ micro micro micro micro λ λ λ micro γ micro

micro micro

micro micro λ λ γ micro

Where in the first inequality we have used the fact that

1

1

11

( )V α +le

and

1

1

bS

micro le in Γ In addition the last

inequality follows from the assumption that 01real le Thus

( ) L t is negative if 0

1real le When 01real lt the derivative

L =

0 if and only if I = 0 while in the case 01real =

the derivative

L = 0 if and only if I = 0 or V = 0 Consequently the



6



largest compact invariant set in (S I V ) isin Γ

L = 0 when 0

1real le is the singleton 0 E Hence LaSallersquos invariance

principle [8] implies that 0 E is globally asymptotically stable in Γ This completes the proof

STABILITY OF THE ENDEMIC EQUILIBRIUM

In this section we discuss the stability of the endemic equilibrium E lowast

Theorem 51 If 01real gt then the endemic equilibrium E lowast of (22) is locally asymptotically stable in

0Γ

Proof At the equilibrium point E lowast

the variational matrix becomes

( )

( )

( )

1 1

1 211

1 11 2

1 1

2

2 2 22

01 1

( ) ( )1 1

0

V S

V V

V S J E r

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+minus minus

+ + = minus + + + +

minus minus +


3 2

1 2 3 0a a aλ λ λ + + + = (51)

Where

11 1 2 2

1

2 01

lowastlowast

lowast= + + + + + ge

+

V a r I

V

λ micro γ micro λ

α

( ) ( )

( )

( )

( )( )

1 12 1 11 2 2

1 1

21 2

22

1

1 11 2 1 1 2

1 1

21 22 1

21

11

1

21 1

1

21 1

1

21

V V a r r I

V V

bS V

V

V V r I r

V V

bS V r

V

V r

V

λ λ micro micro γ γ micro micro λ

α α

λ λ

micro α

λ λ micro γ λ micro γ micro micro

α α

λ λ micro γ micro

micro α

λ micro γ

α

lowast lowast

lowast

lowast lowast

lowastlowast

lowast

lowast lowastlowast

lowast lowast

lowastlowast

lowast

lowast

lowast

= + + + + ++ + +

+ +

minusminus

+

ge + + + + + + + +

+ +

minus+ minus+ +

+

= + + +

+ ( )12 1 1 2

1

[ (22)]1

0

V I fromr

V

λ λ micro γ micro micro

α

lowastlowast

lowast

+ + + + +

+

ge





( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V

λ micro γ micro micro λ

α

micro λ λ

micro α

λ λ λ micro micro λ γ micro

α α

micro λ λ micro micro γ micro

micro α


α α

lowast

lowast

lowast

lowastlowast

lowast

lowast lowast lowastlowast

lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge

It can be easily seen that 1 2 3 0a a a ge By Descartes rule of sign all roots of equation (51) are negative Thus

the endemic equilibrium E lowast

is locally asymptotically stable in0Γ

Now we analyze the global behavior of the endemic equilibrium E lowast

Here we use the geometrical approach of

Li and Muldowney [14] to investigate the global

stability of the endemic equilibrium

E in the feasible region Γ We have

omitted the detailed

introduction of this approach and we refer the interested readers to see [14] We summarize

this

approach below

Let ( ) n x f x Risina be a1C function for x in an open set D sub

n R Consider the differential equation

x = f ( x) (52)

Denote by x (t 0 x ) the solution of (52) such that x(0 0

x ) = 0 x We have following assumptions

( 1 H ) D is simply connected

( 2 H ) There exists a compact absorbing set K sub D

( 3 H ) Equation (52) has unique equilibrium x in D

Let P x a P( x) be a nonsingular (2

n) times (

2

n) matrix-valued function which is

1C in D and a vector norm | middot |

on N

R where N = (2

n) Let micro be the Lozinski˘ı measure with respect to the | middot |

Define a quantity2

q as



8



2 00

1lim sup sup ( ( ( )))

rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f

B P P PJ Pminus minus= + the matrix f P is obtained by replacing each entry p of P by its derivative in the

direction of f ( )ij f p and

[2] J is the second additive compound matrix of the Jacobian matrix J of (52) The following

result has been established in Li and Muldowney [14]

Theorem 52 Suppose that ( 1 H ) ( 2

H ) and ( 3 H ) hold the unique endemic equilibrium

E is globally stable

in0Γ if

2q lt 0

Obviously0Γ is simply connected and

E is unique endemic equilibrium for 0

1real gt in

0Γ To apply the

result of the above theorem for global stability of endemic equilibrium E we first state and prove the following result

Lemma 53 If 01real gt then the system (22) is uniformly persistent in

0Γ

System (22) is said to be uniformly persistent [2] if there exists a constant c gt 0 independent of initial data in

0Γ such that any solution (S (t ) I (t ) V (t )) of (22) satisfies

liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ

The uniform persistence of (22) can be proved by applying a uniform persistence result in [5 Theorem 43] and

using a similar argument as in the proof of proposition 33 of [13] The proof is omitted

The boundedness of0Γ and the above lemma imply that (22) has a compact absorbing set K sub

0Γ [2] Now we

shall prove that the quantity2

q lt 0

Let x = ( ) VS I and f ( x) denote the vector field of (22) The Jacobian matrix J = f

x

part

part associated with a general

solution x(t ) of (22) is

( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +

and its second additive compound matrix[2] J is





11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α

λ λ micro λ micro

micro α

λ γ micro λ micro

α

+ + +minus + + +

minus= minus + + + +

minus + + + + +

Set the function P( x) = P ( ) S I V = diag 1 I I

V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1

f B P P PJ Pminus minus= + can be written in block form as

11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ

γ micro λ micro α

+ + +minus minus

+ = minus minus + + + + +

Consider the norm in3

R as |(u v w)| = max(|u| |v| + |w|) where (u v w) denotes the vector in3

R Let micro denote

the Lozinski˘i measure with respect to this norm Using the method of estimating micro in [15] we have

1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B

21 B are matrix norms with respect to the

1l

vector norm and 10 micro denotes the Lozinski˘i measure with

respect to the1l norm

From system (22) we can write

( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)

Since 11 B is a scalar its Lozinski˘ı measure with respect to any vector norm in

1 R will be equal to 11

B Thus

1110 11 11

1

2( ) 1

V r B B

V

λ micro γ micro α

+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+

On substituting the values of 10 11( ) B micro and

12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I

λ micro γ γ micro

α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=

To calculate 10 22( ) B micro we add the absolute value of the off- diagonal elements

to the diagonal one in each column of 22 B and then take the maximum of two sums see [18] which leads to

( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1

V V I V I V I B r I V I V V I V

λ λ micro λ micro micro γ micro λ micro α α

+ + += minus minus + minus minus + + + + + +





( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V

micro λ micro γ micro λ micro

= minus minus minus minus+ + + + + +

= ( )

1 2 2 I V I I V

micro λ micro minus minus + +


21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro

minus= + minus minus + +

( )

2 1 2 2

V I V I

V I V micro micro λ micro = + + minus minus + +

[From (56)]

( )

1 2

I I

I micro λ = minus +

1 (58)

I

I micro le minus

Using equations (57) and (58) in equation (54) we have

1 2 1( ) sup( ) I

B g g

I

micro micro le le minus (59)

Along each solution 0( ) x t x to (22) such that 0

isin x K the absorbing set we have

( ) 10

1 1 ( )log

(0)le minusint

t I t B dt

t t I micro micro (510)

Which further implies that

2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2

q lt 0 Therefore all the conditions of Theorem (52) are satisfied Hence unique endemic equilibrium

E is

globally stable in0Γ

NUMERICAL SIMULATIONS

In this section we give numerical simulations supporting the theoretical findings When we choose the values of



12



the parameters as 1 200b =

2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the

model then ( 3357049 2472888 2259217) E S I V = = = exists and 0

5128 1real = gt Our simulation shows

endemic equilibrium

E is asymptotically stable (see figure 1) To see the dependence of the steady state value

lowast

I of theinfective population on the parameter lsquor rsquo we have plotted figure 2 for different values of lsquor rsquo keeping all others parameter

values same as for figure 1 and see that the infective population decreases as the parameter lsquo r rsquo increases Further we have

also plotted figure 3 4 5 to see the dependence of steady state value of the susceptible and infective population on the

parameter α 1

keeping all others parameter values same as for figure 1 and noted that the susceptible population increases

and infective population decreases as α 1

increases The details are supplemented by numerical results given in annexure

Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time

Stability graph for the point

(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088 I n f e c t e d H o s t P o p u l a t i o n

Time

Stability graph for I at different values of r

983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

S u s c e p t i b l e H o s t P o p u l a t i o n

Time

Stability graph for S at different values of α1

983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I

n f e c t e d H o s t P o p u l a t i o n

Time

Stability graph for I at different values of α1

983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS

In this paper we have studied a vector-host epidemic model with saturated incidence and a treatment function

proportionate to infectious population I The global stability of the disease-free equilibrium state is established by

Lyapunov method and a geometric approach is used for the global stability of the endemic equilibrium state The basic

reproduction number is obtained and it completely determines the dynamics of the ODE model The model has a globally


a unique positive globally asymptotically stable endemic equilibrium whenever 0real exceeds unity However it is clear

that when the disease is endemic the steady state value

I of the infective individuals decreases as the treatment function

and 1α increases and

I approaches zero as the treatment function and 1

α tends to infinity Thus it will be of great

importance for public health management to maintain the treatment and saturation effects In the absence of the treatment

function and with 10α = the result is perfectly in agreement with Yang et al [7]

Annexure

1b 2

b 1 micro 1 micro γ 1λ 2λ 1α r 0real S

I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I n f e c t e d V e c t o r P o p l a t i o n

Time

Stability graph for V at different values of α1

983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1

C Bowman A B Gumel P van den Driessche J Wu and H Zhu A mathematical model for assessing control

strategies against West Nile virus Bull Math Biol 67 1107ndash1121 2005

2 G Butler H I Freedman and P Waltman Uniformly persistent systems Proceeding of the American

Mathematical Society vol 96 no 3 pp 425ndash430 1986

3

G Cruz-Pacheco L Esteva and J A Montano-Hirose C Vargas Modelling the dynamics West Nile Virus

Bull Math Biol 67 1157ndash1173 2005

4 G Ujjainkar V KGupta B Singh R Khandelwal and N Trivedi An epidemic model with non-monotonic

incidence rate under treatment Applied Mathematical Sciences vol 6 no 24 pp 1159-1171 2010

5

H I Freedman M X Tang and S G Ruan Uniform persistence and flows near a closed positively invariant set

J Dynam Diff Equ 6 583ndash601 1994

6

H Wei X Li and M Martcheva An epidemic model of a vector-borne disease with direct transmission and time

delay J Mathematical Analysis and Applications 2008

7 H Yang H Wei and X Li Global stability of an epidemic model for Vector-borne disease J Syst Sci

Complex vol 23 279-292 2010

8

J P LaSalle The stability of dynamical systems Regional Conference Series in Applied Mathematics SIAM

Philadelphia Pa USA 1976

9

L Cai and X Li Analysis of a simple vector-host epidemic model with direct transmission Discrete Dynamics inNature and Society vol 2010 Article ID 679613 12 pages 2010

10

L Cai S Guo X Li and M Ghosh Global dynamics of a dengue epidemic mathematical model Chaos

Solitons and Fractals vol 42 no 4 pp 2297ndash2304 2009

11 L Esteva and C Vargas Analysis of a dengue disease transmission model Math Biosci 150 131ndash151 1998

12

L M Cai and X Z Li Global analysis of a vector-host epidemic model with nonlinear incidences Applied

Mathematics and Computation vol 217 no 7 pp 3531ndash3541 2010

13 M Y Li J R Graef L Wang and J Karsai Global dynamics of an SEIR model with a varying total population

size Math Biosci 160 191-213 1999

14

M Y Li and J S Muldowney A geometric approach to global-stability problems SIAM Journal on

Mathematical Analysis vol 27 no 4 pp 1070ndash1083 1996

15 R H Martin Jr Logarithmic norms and projections applied to linear differential system J Math Anal Appl 45

392ndash410 1974

16

T K Kar and A Batabyal Modeling and analysis of an epidemic model with non- monotonic incidence rate

under treatment J of Mathematics Research vol 2 no 1 pp 103-115 2010



16



17 T K Kar A Batabyal and RP Agarwal Modeling and analysis of an epidemic model with Classical

Kermack-Mckendrick incidence rate under treatment JKSIAM vol 14 no 1 pp 1-16 2010

18

W A Coppel Stability and asymptotic behavior of differential equations Heath Boston 1965



2



to understand the observed epidemiological patterns disease control and provide understanding of the underlying

mechanisms which influence the spread of disease and may suggest control strategies The model formulation and its

simulation with parameter estimation allow us to test for sensitivity and comparison of conjunctures Treatment plays an

important role to control or decrease the spread of diseases Kar et al [1617] and Ujjainkar et al [4] proposed an epidemicmodel with treatment function Wei et al [6] investigated an epidemic model of a vector-borne disease with direct

transmission and the vector-mediated transmission The global stability of an epidemic model for vector-borne disease was

studied by Yang et al [7] Here we have reanalyzed the model [7] with saturated incidence and a treatment function

proportionate to infectious population I The only mode of transmission in this model is through the vector

The paper is organized as follows In Section 2 a vector-host epidemic model with vector transmissions is

presented where the dynamics of the hosts and vectors are described by SIR and SI model respectively Equilibrium points

and reproduction number are obtained in Section 3 The analysis of stability of the equilibrium of the model is investigated

in Section 4 and Section 5 Numerical analysis and conclusion are given in Section 6 and Section 7

2 THE MATHEMATICAL MODEL

In this section a vector-host epidemic model with vector transmissions is presented and investigated where the

dynamics of the human hosts and vectors are described by SIR and SI model respectively The total host population

1( ) N t is partitioned into three distinct epidemiological subclasses which are susceptible infectious and recovered with

sizes denoted by ( )S t ( ) I t and ( ) R t respectively and the total vector population 2( ) N t is divided into susceptible

and infectious with the sizes denoted by ( ) M t and ( )V t respectively

The dynamics of this infectious disease in the host and vector populations are described by the following system

of nonlinear diff erential equations

1 1

1

1

1

1

2 2 2

2 2

1

1

(1 )

(1 )

dS b S

dt

dI I I rI

dt

dR I R rI

dt

d

SV

V

S

M b MI M dt

dV MI

V

V

V dt

micro α

γ micro α

γ micro

λ micro

λ

λ

λ micro

= minus minus

= minus minus minus

= minus +

= minus minus

= minus

+

+

(21)

With the initial Conditions 0(0)S S = 0

(0) I I = 0(0) R R= 0

(0) M M = and 0(0)V V =

The parameters in the model stand for

1b = Recruitment rate constant in the host population

2b = Recruitment rate constant in the vector population
















which is



N b micro rarrinfin

= and

2 2 2lim t

N b micro rarrinfin




1 1

1

1

1

2

2

1

2

1

2

(1 )

(1

)

SV

V

SV

V

dS b S

dt

dI I I rI

dt

bdV V I V

dt

micro α

γ micro α

micro

λ

λ

λ micro

= minus minus

= minus minus minus

minus= minus

+

+

(22)


Γ = (S I V ) isin3





3 R respectively





4




1

b


E lowast


( ) ( )1 1

1

1

I r V S

V

γ micro α

λ

lowast

lowast + + +

= (31)

( )

2

1 2 1 2 1 2 1

2 1 1 2 1 2 1 2

( )

( )

b b r I

r b b



lowast minus + +=

+ + + +

(32)

( )2 2

2 2 2

b I

V I

λ

micro micro λ

lowastlowast

lowast=

+

(33)


real is



1 2 1 20 2

1 2 1

( )

b b

r

λ λ


+ + (34)








1real lt it is



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( )1 1

0

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

= minus + + + +

minus minus +

V S

V V

V S J r

V V

bV I


1

b






1 11

1

1 10 1

1

2 22

2

0

( ) 0 ( )

0

λ micro

micro

λ γ micro

micro

λ micro

micro

minus minus

= minus + +

minus

b

b J E r

b


( ) 2 1 2 1 21 1 2 2 1

1 2

[ ( ) ( ) ] 0λ λ


+ + + + + + + + minus =bb

r r

( ) ( )21 1 2 2 1 0


(41)








function




( )

( )

( ) ( )

1 2 1 1

2

1 2 1 1 2 21

21

1 1 2 1 21 2 1 2 1 2 1 1 1 21

1 2

1

1

1

2 0 1 2 11

(1

1

0

)

= +




+

le

L I b V

bV I b I V r

b bb I bVI b V r

SV

V

b

V

I V r

λ

λ

micro micro λ



micro micro



1

1

11

( )V α +le

and

1

1

bS





L =


the derivative




6




L = 0 when 0






0Γ



( )

( )

( )

1 1

1 211

1 11 2

1 1

2

2 2 22

01 1

( ) ( )1 1

0

V S

V V

V S J E r

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+minus minus

+ + = minus + + + +

minus minus +


3 2

1 2 3 0a a aλ λ λ + + + = (51)

Where

11 1 2 2

1

2 01

lowastlowast


+

V a r I

V


α

( ) ( )

( )

( )

( )( )

1 12 1 11 2 2

1 1

21 2

22

1

1 11 2 1 1 2

1 1

21 22 1

21

11

1

21 1

1

21 1

1

21

V V a r r I

V V

bS V

V

V V r I r

V V

bS V r

V

V r

V


α α

λ λ

micro α


α α


micro α

λ micro γ

α

lowast lowast

lowast

lowast lowast

lowastlowast

lowast

lowast lowastlowast

lowast lowast

lowastlowast

lowast

lowast

lowast

= + + + + ++ + +

+ +

minusminus

+

ge + + + + + + + +

+ +

minus+ minus+ +

+

= + + +

+ ( )12 1 1 2

1

[ (22)]1

0

V I fromr

V


α

lowastlowast

lowast

+ + + + +

+

ge





( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V


α

micro λ λ

micro α


α α


micro α


α α

lowast

lowast

lowast

lowastlowast

lowast


lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge











this

approach below



x = f ( x) (52)







n) times (

2



on N

R where N = (2


Define a quantity2

q as



8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18

















which is



N b micro rarrinfin

= and

2 2 2lim t

N b micro rarrinfin




1 1

1

1

1

2

2

1

2

1

2

(1 )

(1

)

SV

V

SV

V

dS b S

dt

dI I I rI

dt

bdV V I V

dt

micro α

γ micro α

micro

λ

λ

λ micro

= minus minus

= minus minus minus

minus= minus

+

+

(22)


Γ = (S I V ) isin3





3 R respectively





4




1

b


E lowast


( ) ( )1 1

1

1

I r V S

V

γ micro α

λ

lowast

lowast + + +

= (31)

( )

2

1 2 1 2 1 2 1

2 1 1 2 1 2 1 2

( )

( )

b b r I

r b b



lowast minus + +=

+ + + +

(32)

( )2 2

2 2 2

b I

V I

λ

micro micro λ

lowastlowast

lowast=

+

(33)


real is



1 2 1 20 2

1 2 1

( )

b b

r

λ λ


+ + (34)








1real lt it is



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( )1 1

0

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

= minus + + + +

minus minus +

V S

V V

V S J r

V V

bV I


1

b






1 11

1

1 10 1

1

2 22

2

0

( ) 0 ( )

0

λ micro

micro

λ γ micro

micro

λ micro

micro

minus minus

= minus + +

minus

b

b J E r

b


( ) 2 1 2 1 21 1 2 2 1

1 2

[ ( ) ( ) ] 0λ λ


+ + + + + + + + minus =bb

r r

( ) ( )21 1 2 2 1 0


(41)








function




( )

( )

( ) ( )

1 2 1 1

2

1 2 1 1 2 21

21

1 1 2 1 21 2 1 2 1 2 1 1 1 21

1 2

1

1

1

2 0 1 2 11

(1

1

0

)

= +




+

le

L I b V

bV I b I V r

b bb I bVI b V r

SV

V

b

V

I V r

λ

λ

micro micro λ



micro micro



1

1

11

( )V α +le

and

1

1

bS





L =


the derivative




6




L = 0 when 0






0Γ



( )

( )

( )

1 1

1 211

1 11 2

1 1

2

2 2 22

01 1

( ) ( )1 1

0

V S

V V

V S J E r

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+minus minus

+ + = minus + + + +

minus minus +


3 2

1 2 3 0a a aλ λ λ + + + = (51)

Where

11 1 2 2

1

2 01

lowastlowast


+

V a r I

V


α

( ) ( )

( )

( )

( )( )

1 12 1 11 2 2

1 1

21 2

22

1

1 11 2 1 1 2

1 1

21 22 1

21

11

1

21 1

1

21 1

1

21

V V a r r I

V V

bS V

V

V V r I r

V V

bS V r

V

V r

V


α α

λ λ

micro α


α α


micro α

λ micro γ

α

lowast lowast

lowast

lowast lowast

lowastlowast

lowast

lowast lowastlowast

lowast lowast

lowastlowast

lowast

lowast

lowast

= + + + + ++ + +

+ +

minusminus

+

ge + + + + + + + +

+ +

minus+ minus+ +

+

= + + +

+ ( )12 1 1 2

1

[ (22)]1

0

V I fromr

V


α

lowastlowast

lowast

+ + + + +

+

ge





( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V


α

micro λ λ

micro α


α α


micro α


α α

lowast

lowast

lowast

lowastlowast

lowast


lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge











this

approach below



x = f ( x) (52)







n) times (

2



on N

R where N = (2


Define a quantity2

q as



8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




4




1

b


E lowast


( ) ( )1 1

1

1

I r V S

V

γ micro α

λ

lowast

lowast + + +

= (31)

( )

2

1 2 1 2 1 2 1

2 1 1 2 1 2 1 2

( )

( )

b b r I

r b b



lowast minus + +=

+ + + +

(32)

( )2 2

2 2 2

b I

V I

λ

micro micro λ

lowastlowast

lowast=

+

(33)


real is



1 2 1 20 2

1 2 1

( )

b b

r

λ λ


+ + (34)








1real lt it is



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( )1 1

0

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

= minus + + + +

minus minus +

V S

V V

V S J r

V V

bV I


1

b






1 11

1

1 10 1

1

2 22

2

0

( ) 0 ( )

0

λ micro

micro

λ γ micro

micro

λ micro

micro

minus minus

= minus + +

minus

b

b J E r

b


( ) 2 1 2 1 21 1 2 2 1

1 2

[ ( ) ( ) ] 0λ λ


+ + + + + + + + minus =bb

r r

( ) ( )21 1 2 2 1 0


(41)








function




( )

( )

( ) ( )

1 2 1 1

2

1 2 1 1 2 21

21

1 1 2 1 21 2 1 2 1 2 1 1 1 21

1 2

1

1

1

2 0 1 2 11

(1

1

0

)

= +




+

le

L I b V

bV I b I V r

b bb I bVI b V r

SV

V

b

V

I V r

λ

λ

micro micro λ



micro micro



1

1

11

( )V α +le

and

1

1

bS





L =


the derivative




6




L = 0 when 0






0Γ



( )

( )

( )

1 1

1 211

1 11 2

1 1

2

2 2 22

01 1

( ) ( )1 1

0

V S

V V

V S J E r

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+minus minus

+ + = minus + + + +

minus minus +


3 2

1 2 3 0a a aλ λ λ + + + = (51)

Where

11 1 2 2

1

2 01

lowastlowast


+

V a r I

V


α

( ) ( )

( )

( )

( )( )

1 12 1 11 2 2

1 1

21 2

22

1

1 11 2 1 1 2

1 1

21 22 1

21

11

1

21 1

1

21 1

1

21

V V a r r I

V V

bS V

V

V V r I r

V V

bS V r

V

V r

V


α α

λ λ

micro α


α α


micro α

λ micro γ

α

lowast lowast

lowast

lowast lowast

lowastlowast

lowast

lowast lowastlowast

lowast lowast

lowastlowast

lowast

lowast

lowast

= + + + + ++ + +

+ +

minusminus

+

ge + + + + + + + +

+ +

minus+ minus+ +

+

= + + +

+ ( )12 1 1 2

1

[ (22)]1

0

V I fromr

V


α

lowastlowast

lowast

+ + + + +

+

ge





( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V


α

micro λ λ

micro α


α α


micro α


α α

lowast

lowast

lowast

lowastlowast

lowast


lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge











this

approach below



x = f ( x) (52)







n) times (

2



on N

R where N = (2


Define a quantity2

q as



8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18






1 11

1

1 10 1

1

2 22

2

0

( ) 0 ( )

0

λ micro

micro

λ γ micro

micro

λ micro

micro

minus minus

= minus + +

minus

b

b J E r

b


( ) 2 1 2 1 21 1 2 2 1

1 2

[ ( ) ( ) ] 0λ λ


+ + + + + + + + minus =bb

r r

( ) ( )21 1 2 2 1 0


(41)








function




( )

( )

( ) ( )

1 2 1 1

2

1 2 1 1 2 21

21

1 1 2 1 21 2 1 2 1 2 1 1 1 21

1 2

1

1

1

2 0 1 2 11

(1

1

0

)

= +




+

le

L I b V

bV I b I V r

b bb I bVI b V r

SV

V

b

V

I V r

λ

λ

micro micro λ



micro micro



1

1

11

( )V α +le

and

1

1

bS





L =


the derivative




6




L = 0 when 0






0Γ



( )

( )

( )

1 1

1 211

1 11 2

1 1

2

2 2 22

01 1

( ) ( )1 1

0

V S

V V

V S J E r

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+minus minus

+ + = minus + + + +

minus minus +


3 2

1 2 3 0a a aλ λ λ + + + = (51)

Where

11 1 2 2

1

2 01

lowastlowast


+

V a r I

V


α

( ) ( )

( )

( )

( )( )

1 12 1 11 2 2

1 1

21 2

22

1

1 11 2 1 1 2

1 1

21 22 1

21

11

1

21 1

1

21 1

1

21

V V a r r I

V V

bS V

V

V V r I r

V V

bS V r

V

V r

V


α α

λ λ

micro α


α α


micro α

λ micro γ

α

lowast lowast

lowast

lowast lowast

lowastlowast

lowast

lowast lowastlowast

lowast lowast

lowastlowast

lowast

lowast

lowast

= + + + + ++ + +

+ +

minusminus

+

ge + + + + + + + +

+ +

minus+ minus+ +

+

= + + +

+ ( )12 1 1 2

1

[ (22)]1

0

V I fromr

V


α

lowastlowast

lowast

+ + + + +

+

ge





( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V


α

micro λ λ

micro α


α α


micro α


α α

lowast

lowast

lowast

lowastlowast

lowast


lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge











this

approach below



x = f ( x) (52)







n) times (

2



on N

R where N = (2


Define a quantity2

q as



8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




6




L = 0 when 0






0Γ



( )

( )

( )

1 1

1 211

1 11 2

1 1

2

2 2 22

01 1

( ) ( )1 1

0

V S

V V

V S J E r

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

+minus minus

+ + = minus + + + +

minus minus +


3 2

1 2 3 0a a aλ λ λ + + + = (51)

Where

11 1 2 2

1

2 01

lowastlowast


+

V a r I

V


α

( ) ( )

( )

( )

( )( )

1 12 1 11 2 2

1 1

21 2

22

1

1 11 2 1 1 2

1 1

21 22 1

21

11

1

21 1

1

21 1

1

21

V V a r r I

V V

bS V

V

V V r I r

V V

bS V r

V

V r

V


α α

λ λ

micro α


α α


micro α

λ micro γ

α

lowast lowast

lowast

lowast lowast

lowastlowast

lowast

lowast lowastlowast

lowast lowast

lowastlowast

lowast

lowast

lowast

= + + + + ++ + +

+ +

minusminus

+

ge + + + + + + + +

+ +

minus+ minus+ +

+

= + + +

+ ( )12 1 1 2

1

[ (22)]1

0

V I fromr

V


α

lowastlowast

lowast

+ + + + +

+

ge





( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V


α

micro λ λ

micro α


α α


micro α


α α

lowast

lowast

lowast

lowastlowast

lowast


lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge











this

approach below



x = f ( x) (52)







n) times (

2



on N

R where N = (2


Define a quantity2

q as



8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18






( )( )

( )

( )( ) ( )

( )( )

( )( ) ( )

13 1 1 2 2

1

21 1 2

22

1

1 2 1 21 21

1 1

21 1 21 2 1

21

1 2 1 21 21

1 1

1

1

1 1

1

1 1

V a r I

V

bS

V V

V I V I r

V V

bS V r

V

V I V I r

V V


α

micro λ λ

micro α


α α


micro α


α α

lowast

lowast

lowast

lowastlowast

lowast


lowast lowast

lowastlowast

lowast


lowast lowast

= + + + +

+

minusminus +

ge + ++ +

+ +

minus+ minus+ +

+

= + ++ +

+ + [ (22)]

0

from

ge











this

approach below



x = f ( x) (52)







n) times (

2



on N

R where N = (2


Define a quantity2

q as



8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




8



2 00


rarrinfin rarr

= intot x K

t q B x s x ds

t micro (53)

Where1 [2] 1

f








in0Γ if

2q lt 0



1real gt in

0Γ To apply the



0Γ



liminf t

S crarrinfin

ge liminf t

I crarrinfin

ge liminf t

V crarrinfin

ge

Provided (S (0) I (0) V (0)) 0

isin Γ




0Γ [2] Now we


q lt 0


x

part



( )

( )

( )

1 11 2

1 1

1 11 2

1 1

2

2 2 2

2

01 1

( ) 1 1

0

V S

V V

V S r J

V V

bV I

λ λ micro

α α

λ λ γ micro

α α

λ λ micro micro

+minus minus + +

minus + +=

+ +

minus minus +






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18






11

[2 ]

22 23 13

32 11 33 12

31 21 22 33

a a a a

J a a a a

a a a a

+ minus

= + minus +

( ) ( )

( )

1 1 11 2 2

1 1 1

[2] 2 12 1 2 2

2 1

11 2 2

1

21 1 1

( ) 0 1

01

V S S r

V V V

b V V J I

V

V r I

V

λ λ λ micro γ

α α α


micro α


α

+ + +minus + + +


minus + + + + +


V V

then

1

0 f

I V I V P P diag

I V I V

minus = minus minus

The matrix

1 [2] 1


11 12

21 22

B B B

B B

=

With1

111

1

21

V r B

V

λ micro γ

α

+ + += minus +

( ) ( )1 1

12 2 2

1 1

1 1

SV SV B

I I V V

λ λ

α α

=

+ +

22

221

0

b I V

B V

λ

micro

minus

=

( )

1

1 2 2

1

22

11 2 2

1

0

1

1

V I V I

V I V BV I V

r I V I V

λ micro λ micro

α λ


+ + +minus minus




R Let micro denote


1 2( ) sup( ) B g g micro le (54)

Where1 10 11 12

( )g B B micro = + 2 10 22 21

( )g B B micro = +



10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




10



12 B


1l




( )

1

1

1

)

(1 I

SV

V

I r

I micro

α

λ γ = minus + +

+ (55)

2

2 2

2

bV I V

V V λ micro

micro

minus= minus

(56)



B Thus

1110 11 11

1

2( ) 1

V r B B

V


+ + += = minus +

Also

( )1

12 2

1

1

SV B

I V

λ

α =

+


12 B 1

g will become

( )

1 111 2

1 1

2

1 1

V SV r g

V I V

λ λ micro γ

α α

+ + += minus +

+ +

( )1 1

1

1 1

21 1

V SV r

V I V

λ λ micro γ

α α

+ + +le minus + + +

( )

11 1

1

21

V I r r

V I


α

+ + += minus + + + + +

[From (55)]

1

1

11

λ micro

α

+= minus

+

V I

V I

1 ( 5 7 ) I

I micro le minus

Now22

21

2

b I

V BV

λ

micro

minus=



( )

1 11 2 210 22 1 2 2

1 1

( ) sup 1 1








( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18






( ) ( )

1 2 2 1 2 2sup I V I V

I r I I V I V



= ( )

1 2 2 I V I I V



21 B 2

g will becomes

( )

222 1 2 2

2

b I I V V g I

V I V

λ micro λ micro

micro


( )

2 1 2 2

V I V I


[From (56)]

( )

1 2

I I


1 (58)

I

I micro le minus


1 2 1( ) sup( ) I

B g g

I




( ) 10

1 1 ( )log

(0)le minusint

t I t B dt



2 1

1 ( )limsup sup log

(0)ot x K

I t q

t I micro

rarrinfin rarr

le minus

1 micro = minus

0lt

ie2


E is






12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




12




2 300b =1 05 micro =

2 06 micro = 04γ = 1001λ = 2

002λ = 101α = 04r = for the



endemic equilibrium


lowast




parameter α 1




Figure 1

Figure 2

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983088 983093983088 983089983088983088 983089983093983088 983090983088983088 983090983093983088

P o p u l a t i o n

Time


(S=3357094 I=2472888 V=2259217)

983123

983113

983126

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088983091983093

983092983088


Time


983113 983137983156 983154983101983088

983113 983137983156 983154983101983088983086983090

983113 983137983156 983154983101983088983086983092

983113 983137983156 983154 983101 983088983086983094

983113 983137983156 983154 983101 983088983086983096

983113 983137983156 983154 983101 983089





Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18






Figure 3

Figure 4

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983091983093983088

983092983088983088

983092983093983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983123 983080983088983086983089983081

983123 983080983088983086983090983081

983123 983080983088983086983091983081

983123 983080983088983086983092983081

983123 983080983088983086983093983081

983088

983093

983089983088

983089983093

983090983088

983090983093

983091983088

983091983093

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088

I


Time


983113 983080983088983086983089983081

983113 983080983088983086983090983081

983113 983080983088983086983091983081

983113 983080983088983086983092983081

983113 983080983088983086983093983081



14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




14



Figure 5

CONCLUSIONS














Annexure

1b 2


I

V

200 300 05 06 04 001 002 01 0 7407407 3353115 3593807 2725138

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 01 04 5128205 3357049 2472888 2259217

200 300 05 06 04 001 002 01 06 4444444 3359016 2136612 2079787

200 300 05 06 04 001 002 01 08 3921569 3360984 187946 1925889

200 300 05 06 04 001 002 01 1 3508772 3362951 1676445 1792435

200 300 05 06 04 001 002 01 02 6060606 3355082 2931446 2471106

200 300 05 06 04 001 002 02 02 6060606 3645586 1610975 1746892

200 300 05 06 O4 001 002 03 02 6060606 3755652 1110672 1350962

200 300 05 06 04 001 002 04 02 6060606 3813555 8474795 1101344200 300 05 06 04 001 002 05 02 6060606 3849272 6851271 9295841

983088

983093983088

983089983088983088

983089983093983088

983090983088983088

983090983093983088

983091983088983088

983088 983090983088 983092983088 983094983088 983096983088 983089983088983088 983089983090983088


Time


983126 983080983088983086983089983081

983126 983080983088983086983090983081

983126 983080983088983086983091983081

983126 983080983088983086983092983081

983126 983080983088983086983093983081





REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18






REFERENCES

1





3





5



6




Complex vol 23 279-292 2010

8



9


10




12





14




392ndash410 1974

16





16





18




16





18


Documents

1. IJAMSS - Modeling and Analysis of a Vector-Host Epidemic Model - Sneha Porwal - OPaid