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International Journal of Applied and Natural Sciences (IJANS) ISSN(P): 2319-4014; ISSN(E): 2319-4022 Vol. 3, Issue 1, Jan 2014, 33-40 © IASET

CRITICAL ANALYSIS ON BIFURCATION OF SPR_SODE MODEL FOR THE SPREAD OF DENGUE S. DHEVARAJAN1, A. IYEMPERUMAL2, S. P. RAJAGOPALAN3 & D. KALPANA4 1,2,3

Dr. MGR Educational and Research Institute University, Chennai, Tamil Nadu, India 4

PSB Polytechnic College, Chennai, Tamil Nadu, India

ABSTRACT A ordinary differential equation with stochastic parameters, called SPR_SODE model for the spread of dengue fever is considered. Critical Values of bifurcation and the boundary of the above said model are discussed. In this paper, the different parameters are considered for further analysis. The bifurcation at the characteristic value of the non-linear eigen value equation is supercritical if 1  0 and subcritical if 1  0 . The equilibrium solution pair in the positive octant of

R 7 is also discussed.

KEYWORDS: Boundary, Critical Values, ODE, SPR_SODE Model, Stability INTRODUCTION Dengue is a disease comes under infectious diseases which is in worldwide. The work of a carrier (i.e) the medium for transmitting is performed by the mosquito, “Aedes Ageypti [7]. There are so many models for such infectious diseases. We need a separate model for such a special disease like dengue fever for better results. The asymptotically stable equilibrium points or equilibrium solutions can be defined as the equilibrium solutions in which solutions that start “near” them move toward the equilibrium solution [1]. In this work, the SPR_SODE model [2], [3], [4] (SPR_Stochastic Ordinary differential Equation model) is considered to analyze further. All the notions of SPR_SODE model [2], [3],[4] are taken for further analysis without any change and the same model is given below.

SPR_SODE MODEL d  SS    b  TP    L   RC    (t )  SS    TP    SS  h h   h   h h    h   h    h    h  h dt  h  d  EX    (t )  SS     EX    TP    EX  h    h   h dt  h  h   h h   h d  IF     EX    RC   IF    TP    RC  h    h    h dt  h  h   h   h   h d  RC     IF   L  RC    TP    IF    IF  h    h   h  h dt  h  h   h h   h d  SS   [ BIR] TP   (t )  SS     TP    SS  m m m  m  m     m  m dt  m   d  EX    (t )  SS     EX     TP    EX  m   m   m dt  m  m   m m   m 


S. Dhevarajan, A. Iyemperumal, S. P. Rajagopalan & D. Kalpana

34

d  IF     EX    TP    IF  m   m   m dt  m  m   m 

(A)

By converting (A) to fractional quantities and denoting each scaled population by small letters, one can get, [2], [3], [4]   h   . TP  . 1  [ex]h  [if ]h  [re]h   h  [ BIR]h   [ex]h  h[if ]h[ex]h    m  TP    h  h  

d ex   hm Pmh[if ]m dt   h m TP    TP  

m

h

 2    if  h if  h h  h 

d if    ex      BIR   h  h TP  dt   h h   h  h  

    rc  h h if  TP  h h  h 

d rc    if    L  [ BIR] h h dt   h h   h  h TP  

d TP     TP   [DID]  [ DDD] TP   . TP   [if ] [TP] h h h h h h h  h   h   h dt   h

hm d ex   . TP  .  P [if ]  P [rc]  . 1  [ex]h  [if ]h   h  [ BIR]m [ex]m dt   m m TP    TP    h  mh h mh h   h    m

h

d if    ex [BIR] if  m   m dt   m m   m d TP   [ BIR] TP   [ DID]  [ DDD] TP   TP m m m  m  m   m dt   m 

(B)

Now, R0 can be defined as, R0  hm .mh , [2],[3],[4]

(C)

where hm and mh can be written in mathematical notation as, hm 

mh 

hm Pmh TP  h *

m

*

*

hm Pmh TP  m *

h

*

1

   [DID]  [DDD] TP  *

h  [ DID]h  [ DDD]h TP  h h TP  h  m TP  m *

Pmh [ DID]m  [ DDD]m TP  m  *

m  [ DID]m  [ DDD]m TP  m m TP  m  h TP  h *

*

 * .  Pmh  Pmh .h h  [ DID]m  [ DDD]m TP  h 

h

h

h

h

1

h

1 

.

(D)

 

In [2,3,4, S. Dhevarajan et.al, 2013], it was proved that the above said model is asymptotically stable. It is also proved that the domain, existence and uniqueness of the solution of the above said model. The disease free equilibrium of the model is also proved [3]. The solution of the above said model is asymptotically stable[2].

The disease free

equilibrium solution is also exists [5]. Asymptotically equilibrium points are near to equilibrium points that are near to them move toward the equilibrium solution [1]. The Existence of Endemic Equilibrium Points of SPR-SODE Model The equilibrium equations for (B) are shown below in (E). In this analysis, Here, the terms


Critical Analysis on Bifurcation of SPR_SODE Model for the Spread of Dengue

ex , if  , rc , TPh , ex , if  h  h h m  m

35

and TPm are representing their respective equilibrium values and not their

actual values at a given time t. 0

hm Pmh[if ]m  m TP   h TP  m

0  h ex 

  h   . TP  . 1  [ex]h  [if ]h  [re]h   h  [ BIR]h   [ex]h h[if ]h[ex]h   m    TP     h h  

 

    BIR   h  h  h

 

h TP   

    if  h if h  h

2   h

(E1)

(E2)

  h     0  h if    Lh  [ BIR]h  rc  h if  TP  h h  h TP     h  h  

(E3)

  0 h  h TP   [ DID]h  [ DDD]h TP   . TP  h[if ]h[TP]h  h   h   h

(E4)

0

hm . TP  .  P [if ]  P [rc]  . 1  [ex]h  [if ]h   h  [ BIR]m [ex]m m TP   h TP    h  mh h mh h   m

0  m ex

m

(E5)

h

[BIR]m if 

(E6)

m

0  [BIR]m TPm  [DID]m  [DDD]m TP m TP m

(E7)

Define new parameter,   h / m , to obtain * *        h   TP m  TP h     TP Pmh .[if ]m 1 [ex]h  [if ]h  [re]h   h  [ BIR]h  [ex]h h[if ]h[ex]h  0 m    TP   TP  TP         m h h   

(E8)

* *       TP  m  TP  h    . TP  .  Pmh[if ]h  Pmh[rc]h  . 1 [ex]h [if ]h   h  [ BIR]m [ex]m  0 h        TP   TP    m   h   

(E9)

The bifurcation parameter  can be varied, while keeping all other parameters fixed. In terms of the original variables, this corresponds to changing h and m , while keeping the ratio between them fixed. Consider   h / m . One can choose the ratio  to sweep out the entire parameter space. Hence,

TPm 

[ BIR]m  [ DID]m [ DDD]m

 m ex  m Solving for if  in (E6) in terms of ex , one can find if    m  m m [ BIR ]m

(E10)

(E11)

By rewriting the positive equilibrium for TP h in terms of if  from (E9) as  h

[ BIR]h  [ DID]h  [if ]   [ BIR]h  [ DID]h  [if ]   4[ DDD]hh h h h h    2[ DDD]h 2

TP h

(E12)


S. Dhevarajan, A. Iyemperumal, S. P. Rajagopalan & D. Kalpana

36

Using (E12) in (E3), and solving for rc  in terms of if  :  h h

rc   h

2h if  h 2Lh  [ BIR]h  [ DID]h h[if ]h 

[ BIR]

(E13)

 [ DID]h  h [if ]h   4[ DDD]hh 2

h

It is given the nonlinear nature of (E2), which is not possible to solve for if  in terms of  h Now,

g if 

by

h



using

h  

h

(E12) 1 2

[ BIR]

h

rewrite

(E2),

and

define

the

function ex   g if  h  h

 [ DID ]h   h [if ]h   [ BIR ]h  [ DID ]h h [if ]h   4[ DDD ]hh

It is clear that g (0) = 0. Coin the positive constant g (1) as exmax  . As g if   h h

continuous function of if  with g ' if  > 0 for if  ∈ [0, 1] and  h  h  h

y ex h

as

2

h

smooth function if  = y ex  h  h

ex explicitly. h

if   h

 is a nice smooth and

ex ∈ 0, exmax   , there exists a h  h

with domain 0, exmax   and range [0, 1]. As g (0) > 0, the smooth function h 

would extend to some small

ex by substituting if  = y ex  h  h h

TP h

ex < 0. h

and rc  can also be expressed as functions of h

into (E12) and (E13). Now introduce D1 , the bounded open subset of

R2 defined by

  D1    

 ex   ex   ex  ex     h  2   h   h  max  h  ex   ex   ex  1 m m   m 

    

(E14)

for some ex  > 0 and some ex > 0. Define Z as the open and bounded set Z = {  ∈ R|−MZ < h m

 <MZ}.

This set is defined to include the characteristic values of L, so there is minimum value that MZ can have, but MZ may be arbitrarily large with  

mh . m N m*  h N h*

Determination of Lower Order Terms Now it is to determine lower order terms. (E2) can be written as f

 ex

f

 ex

, if  h

h 

= 0, where

, if  = h

h 

1   h ex   h  h  [ BIR]h  [ DID]h  h [if ]h   [ BIR]h  [ DID]h h[if ]h   4[ DDD]hh  if  h  2   h 2

and use implicit differentiation to write if  = y ex  as a Taylor polynomial of the form h  h

if   1 ex   O ex   h  h  h

2

(E15)


Critical Analysis on Bifurcation of SPR_SODE Model for the Spread of Dengue

 f    ex  Where 1     h  f   if    h

37

   h   1 2    h  h  2 [ BIR]h  [ DID]h   [ BIR]h  [ DID]h   4[ DDD]hh   exh if h  0

The first order approximations to the equilibrium equations can be obtained by substituting (E15) into (E13) and (E12), and then all three, along with (E11) and (E10) into the equilibrium equations (E8) and (E9). Hence,

 0   f1_10   f  0   2 _10

 ex   f1_ 01   ex  h      h   O   ex   f 2 _ 01   ex     m    m 

2

(E16)

1  Where, f1_10   h  h  [ BIR]h  [ DID]h   2  f1_ 01  

[ BIR]h  [ DID]h 

2

  4[ DDD]hh  

(E17a)

 m Phm .[ BIR]m  [ DID]m 

f 2 _10  .

(E17b)

[ BIR]m [ DDD]m  h [ BIR]h  [ DID]h   

[ BIR]h  [ DID]h 

2

 4[ DDD]hh  

  1 2 2[ DDD]h  h  h  [ BIR]h  [ DID]h    [ BIR]h  [ DID]h   4[ DDD]hh   2        h . P mh P    mh    1 2   Lh  [ BIR]h  [ DID]h    [ BIR]h  [ DID]h   4[ DDD]hh  2    

f 2 _10   [ BIR]m   m 

(E17c)

(E17d)

To apply Corollary 1.12 of Rabinowitz [6], we algebraically manipulate (E16) to produce

u  Lu  h(, u )

(E18)

 ex   0  h  Where u   and L    ex   B   m  B

A  With 0

 h [ BIR]h  [ DID]h   

(E19a)

[ BIR]h  [ DID]h 

2

  1 2[ DDD]h [ BIR]m   h   h   h  [ BIR]h  [ DID]h    2     P   mh    Lh  

 4[ DDD]hh  

[ BIR]h  [ DID]h 

2

 h . P mh   1 [ BIR]h  [ DID]h    2 

[ BIR]h  [ DID]h 

2

  4[ DDD]hh  

(E.19b)

     4[ DDD]hh  

and h(  , u) is O(u2). The two characteristic values of L by 1

 1/ AB and 2  1/ AB . As both A and B are

always positive, due to our assumption that  BIR  m >  DID m ,

1 is real and corresponds to the dominant eigen value of

L. The right and left eigen vectors corresponding to

1 are respectively,


S. Dhevarajan, A. Iyemperumal, S. P. Rajagopalan & D. Kalpana

38

v

T

A

B

w

and

For MZ > 1 , as 0 ∈ Y, ( 1 , 0)

B

A

(E20)

 . We denote the continuum of solution-pairs emanating from ( 1 , 0) by

1 , where 1  , and from ( 2 , 0) by  2 , where  2  . Let Z1 , Z 2 ,U1 and U 2 are the sets defined by,

Z1    Z u such that  , u  1

(E21a)

U1  u  Y  such that  , u  1

(E21b)

Z2   Z u such that  , u  2 

(E21c)

U 2  u  Y  such that  , u  2 

(E21d) [Change the variable, Y and Z]

The by Y

+



part

of

Y

in

the

quadrant

of

R2

can

be

denoted

by

Y+

and

defined

ex  , ex  Y h m

 ex     h  Y +   Y  ex  m  

positive

+ ex h >0 and ex m > 0 . The internal boundary of Y can be defined by

 ex  >0 and   h  ex  = 0   m

  ex  =0 and  or    h   ex  >0    m

  ex  =0 and  or    h   ex =0    m

    

Now, it is to expand the terms of the nonlinear eigen value equation (E18) about the bifurcation point, ( 1 , 0). The expanded variables are

u  0   u(1) + 2u(2) +... ….. (E22a) ;

  1  1 + 2 2 +...

h  , u  ��� h  1  1 + 2  2 +... , u (1) + 2u (2) +...   2 h2  1 , u (1)   ....

Now, consider,

1  

w.h2 w.Lv

(E22b)

(E22c)

(E23)

Now it is to prove that if v and w are the right and left eigenvectors of L corresponding to the characteristic value

1 , respectively, the bifurcation at  = 1 of the nonlinear eigen value equation (E18) is supercritical if 1 > 0 and

subcritical if 1 < 0. Evaluating the substitution of the expansions (E22) into the eigen value equation (E18) at O(ε2), we obtain u(2) =

1 Lu(2) + 1 Lu(1) + h2, which we can rewrite as (I −

1 L) u (2) = 1 Lv + h2,

Where I is the 2 × 2 identity matrix. As

(E24)

1 is a characteristic value of L, (I − 1 L) is a singular matrix. Thus, for

(E24) to have a solution, 1 Lv + h2 must be in the range of (I − 1 L); i.e., it must be orthogonal to the null space of the


Critical Analysis on Bifurcation of SPR_SODE Model for the Spread of Dengue

39

adjoint of (I− 1 L). The null space of the adjoint of (I− 1 L) is spanned by the left eigenvector of L (corresponding to the eigen value 1/ 1 ), w (E20). The Fredholm condition for the solvability of (A.24) is w. ( 1 Lv + h2) = 0. Solving for

1 provides (E23) obviously. If 1 is positive, then for small positive ε, u > 0 and  > 1 , and the bifurcation is supercritical. Similarly, if 1 is negative, then for small positive ε, u > 0 and Now, it is to prove that For all u  U1,

 < 1 , and the bifurcation is subcritical.

ex > 0 and ex > 0. m h

From [5], there are no equilibrium points on ∂Y + other than

ex = ex = 0, so U1 ∩ ∂Y + = 0. From [5], m h

close to the bifurcation point ( 1 , 0), the direction of U1 is equal to v, the right eigenvector corresponding to the characteristic value, u  U1,

1 . As v contains only positive terms, U1 is entirely contained in Y +. Hence, for all

ex > 0 and ex > 0. m h

Equilibrium in the Boundary of the Positive Orthant Now it is to prove that the point u = 0  Y corresponds to

xnodis

∈ R7 (on the boundary of the positive

orthant of R7). For every solution-pair (  , u)  1 , there corresponds one equilibrium-pair (  , x )  Z × R7, where *

x*

is in the positive orthant of R7. First it is to show that u = 0 corresponds to xnodis . As

ex = ex = 0, From [1] & [6], One can argue that m h

the only possible equilibrium point is xnodis . Now it is to show that for every

  Z1 there exists an x * in the positive

orthant of R7 for the corresponding u  U1. (E25) implies ex > 0 and ex > 0. The equilibrium equation (E11) m h implies for every positive ex , there exists a positive if  . The equilibrium equation for  m m

TP m

has a positive and

bounded solution, depending only on parameter values (E10). Now, if  = y ex  implies for every positive h  h

ex  there exists a positive if  . The equilibrium equations (E13) and (E12) show that, for every positive if   h  h h there exists a positive rc  and h

TPh . Hence, the point u = 0  Y corresponds to

xnodis  R7 (on the boundary of the

positive orthant of R7). For every solution-pair (  , u)  1 , there corresponds one equilibrium-pair (  , x )  Z × R7, *

where

x * is in the positive orthant of R7…….(E26)

CONCLUSIONS The different parameters of the SPR_SODE model are considered for further analysis. Here it is proved that the bifurcation at the characteristic value of the non-linear eigen value equation is supercritical if 1  0 and subcritical if

1  0 . TPh . It is also proved that the point u = 0  Y corresponds to xnodis  R7 , on the boundary of the positive orthant of R7) and For every solution-pair (  , u)  1 , there corresponds one equilibrium-pair (  , x )  Z × R7, where *

x * is in the positive orthant of R7.


S. Dhevarajan, A. Iyemperumal, S. P. Rajagopalan & D. Kalpana

40

REFERENCES 1.

Boyd, John P. "The Devil's Invention: Asymptotic, Super asymptotic and Hyper asymptotic Series". Acta Applicandae Mathematicae 56 (1): 1–98, March 1999.

2.

Dhevarajan. S, Iyemperumal. S, Rajagopalan. S.P. and Kalpana.D, “Asymptotic Stability of SPR_SODE Model for Dengue”, International Journal of Research in Applied, Natural and Social Sciences, ISSN (E): 2321-8851; Vol. 1, Issue 6, 59-64, 2013.

3.

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4.

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5 applied ijans critical analysis on bifurcation s dhevarajan