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AnExtension

of SIRS

Model


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Model


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Gajendra UjjainkarB. Singh

V. K. GuptaR. Khandelwal &

Neetu Trivedi


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Abstract


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Pathak et.al. [11] discussedrich dynamics of an SIRS modelwith an asymptotichomogeneous transmission

function. We have extendedthis model for a more realisticsituation. Stability criteria for

disease free as well as endemicsituation are also discussed.Numerical simulations are alsoerformed for critical


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Introduc

tion


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History of mathematical epidemiology and basics ofSIR epidemic models may be found in Bailey [8],Murray [6], and Anderson and May [10]. The study of

the disease transmission in epidemiology is mainlyconcern with the rate of the transmission of thedisease and the stability. The rate of transmission orthe incidence rate has been widely used and studiedin many different kinds of models. Rate of incidence

is the rate at which the susceptible populationbecomes infectious. It is always possible that afterbeing infected, only a part of that population isinfectious and responsible for the spread of further

infection. The model we are studying here is the SIRSepidemic model. The transmission functionconsidered by Diekmann and Kretzschmar [9], andZegeling and Kooij [1] has been generalized in Pathaket al. [11]. We have considered the same saturation

effect with a reduction in infectious part. Numerical


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The MathematicalModel


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th ep ro p o s e d m o d e li sd S k S I

b d S Rd t 1 S I

= + + +

(1)

d I k S I( d ) I

d t 1 S I

= +

+ +

(2)

d RI (d ) R

d t= + (3)


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Equilibrium

pointsand

stability


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k S Ib d S R 0

1 S I + =

+ + (4)

k S I( d ) I 0

1 S I

+ =

+ + (5)

I ( d ) R 0 + = (6)


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By equation (6) we have IR(d )

=+

and by (5)

kS(d )

(1 S I)

= +

+ +

or

(d )(1 I)S

[ k (d )]

+ + =

+ ,

Adding (5) and (6) we get

(d )b dS I R 0+ + =


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P u t th e v a l u e s o f S a n d R i n ( 4 )t o g e t

( d )(1 I ) ( d ) Ib d I 0[ k ( d )] ( d )

+ + + + = + +

A f te r si m p l i fi c a ti o n w e ge t ,

[ b k ( d )( b d )]I

d (d )( d ) { k ( d )} {d ( d ) (1 )}

+ +=

+ + + + + + +

D e f in e t h e b a s ic re p r o d u c t io n n u m b e r a s

0

b k b (d )R

d ( d )

+ =

+


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T h e o r e m 1 .(i)if R0 1, then there is no po si t ive eq ui l ibrium ;

(ii)if R0 > 1, then the re is a u niqu e p osi t ive eq ui l ibriumE*(S* , I*, R*) o f th e sy st em

which is theendemic equi l ibr iumgiven by

.

(d )(1 I*)S*

[ k (d )]

+ + =

+

[ b k (d )(b d )]I*d (d )(d ) { k (d )}{d (d ) (1 )}

+ += + + + + + + +

A n d

I *R *

(d )

=

+


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N o w to c h e c k th e s ta b il i ty o f th e e q u il ib r iu m p in ts we f ind theJ a c o b i a n m a t rix

2 2

2 2

(1 I ) k I (1 S ) k Sd

(1 S I ) (1 S I )

(1 I ) k I (1 S ) k SJ (d ) 0

(1 S I ) (1 S I )

0 ( d )

+ + + + + + + + = +

+ + + + +

.


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A t the disease free equilibrium point, the jacobian is

0

k b / dd

(1 b / d )

k b / dJ 0 (d ) 0

(1 b / d )

0 (d )

+

= + +

+

H ence the point (b/d,0, 0) is stableif

k b / d(d )

(1 b / d )

+

+


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Now the jacobian at the endemic point E*(S*, I*, R*) is

2 2

2 2

(1 I*)kI* (1 S*)kS*d

(1 S* I*) (1 S* I*)

(1 I*) kI* (1 S*) kS*J* (d ) 0

(1 S* I*) (1 S* I*)

0 (d )

+ + + + + + + +

= + + + + +

+


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W e n o t e th a t loo k i n g t o th e v a l u e o f th e t r a c e a n d d e t e r m i n a n t o f th e j a c o b i a n, t h e f o l lo w i n gc a s e s

a r e o b t a in e d :

1 . If th e T r a c e ( J * ) < 0 a n d D e t (J* ) > 0 , t h e n t h e p o i n t is a s t a b l e n o d e o r a s t a b l e s p i ra l .

2 . If th e T r a c e (J* ) > 0 a n d D e t (J* ) > 0 , th e n t h e p o i n t is a n u n st a b le n o d e o r u n s t a b l e s p i ra l .

3 . I f t h e D e t (J* ) < 0 , th e n t h e p o i n t is a s a d d l e p o i n t.

4 . I f t h e T r a c e(J* ) = 0 , th en the re a r e s om e l im i t cyc le s .

T h u sw iths u i ta b l e v a l u e so f th e p a r a m e t e rs w h e r e w e g e t e q u i li b r iu m p o i n t , t h e v a l u e

d e t e r m i n an t g i v e t h e i d e a o f t h e s t a b i li ty c o n d i t io n s .


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Numericals


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Consider the following values:

d 2.29, 3.1, 4.7, b 3.1, 1.5, k 9, 0.19= = = = = = =

Here we note that for the values of from 0.1 to 0.854, we get negativevalue of R0. From =0.855

to 1 we have R0 < 1. In this case there exists no positive equilibrium point.Though the values ofI

and R (at = 1) given in [11] are considered to be zero, as they are near to zero,and the point is

considered as a disease free equilibrium point. We tabulatehere the exact values.


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R0 S I R

0.86 0.028384279 1.60586789 -0.20560182 -0.01030722

0.88 0.126637555 1.57110796 -0.18149476 -0.00909868

0.9 0.22489083 1.53907197 -0.15836499 -0.00793914

0.92 0.323144105 1.5094536 -0.13610188 -0.00682305

0.94 0.42139738 1.48199077 -0.11461077 -0.00574566

0.96 0.519650655 1.45645797 -0.09381023 -0.00470289

0.98 0.61790393 1.4326601 -0.07362978 -0.0036912

1 0.716157205 1.41042749 -0.05400816 -0.00270753


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N o w w e t a k e t h e v a l u e s o f p a r a m e t e r s a s f o l l o w s :

d 0 . 2 9 , 3 . 1 , 4 . 7 , b 3 .1, 1 .5 , k 6 . 5 , 0 . 1 9= = = = = =

T h e v a lu e o f r a n g e s f ro m 0 . 1 t o 1 . 0. At 0 . 1 a n d0 .2 w eg e t R 0 < 1 a n dd o n o t g e t p o s

e q u i l i b r i u m p o i n t. We t a b u l a te h e r e t h e e q u i l i b r iu m p o i n t s w i t h = 0 . 3 t o1 .0


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R0 S I R

0.3 10.2887931 5.82262600 0.979633517 0.103983446

0.4 24.76436782 4.136243065

1.826020518 0.193823407

0.5 39.23994253 3.57249213 2.577451746 0.273584264

0.6 53.71551724 3.31900805 3.335747969 0.354073807

0.7 68.19109195 3.19732076 4.126860015 0.438046594

0.8 82.66666667 3.14578340 4.963274822 0.526828054

0.9 97.14224138 3.13769135 5.853993656 0.621373628

1 111.6178161 3.1598476 6.807250221 0.722557286


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W ith = 0 .3d 0 .2 9 , 3 .1, 4 .7 , b 3 .1, 1 .5 , k 6 .5 , 0 .1 9= = = = = =,

A s g iv e n ta b le s h o w s , w e g e t e q u i l ib r iu m p o i n t

(5 .8 2 2 6 2 6 0 0 5 , 0 .9 7 9 6 3 3 5 1 7 , 0 .1 0 3 9 8 3 4 4 6)

w h i c h i s s ta b l e a s s e e n i n th e f o l lo w in g g r a p h


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CONCLUS

ION


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T h e a n a l y s i s o f t h e m o d e l s t u d i e s m o r e c l o s e l y t h e e f

p r o v i d e s t h e g e n e r a l i z a t i o n o f t h e m o d e l g i v e n i n [1 1] i n S I RS c o m p a r t m e n t s a n d t h e

r e s u l t g i v e n i n [1 1] c a n e a s i l y b e v e r i f i e d a s a p e r t i c a l a r c a s e t a k i n


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REFEREN

CES


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1. A. Zegeling, R.E. Kooij, Uniqueness of limit cycles in models for micro parasitic and

Macro parasitic diseases,J. Math. Biol., 36, (1998). 407417,

2. F. Brauer, Some Simple Epidemic Models, Mathematical biology and engineering3(1)

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3. H.W. Hethcote and P. Van Den Driessche: Some Epidemiological Models with Non-linear

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8. N.T.J. Bailey, The Mathematical theory of Infectious Diseases, Griffin, London, (1975)


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9. O. Diekmann, M. Kretszscmar,Patterns in the effects of infectious disease on population

growth, J. Math. Bio, 29( 1991) 539-570

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Behavior of SIRS Epidemiological Models. J. Math. Biol.(1986) 187-204.


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