Mathematical Model of Dengue Fever with and without awareness in Host Population

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1 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato ISSN: , October 015 Mathematcal Model of Degue Fever wth ad wthout awaree Hot Populato Gaga Ram Phajoo 1* & Dl Bahadur Gurug 1 Departmet of Natural Scece (Mathematc), School of Scece Kathmadu Uverty, Dhulkhel, NEPAL *E-mal: gagaram@ku.edu.p Departmet of Natural Scece (Mathematc), School of Scece Kathmadu Uverty, Dhulkhel, NEPAL E-mal: db_gurug@ku.edu.p Abtract: Degue oe of the mot rapdly preadg deae the world. It tramtted to huma by the bte of fected aede moqutoe. I the preet paper, the tramo dyamc of degue deae the preece ad abece of awaree hot populato dcued. It aumed that ome hot do ot teract wth the fected moqutoe a they take dfferet kd of precauto due to ther awaree toward the deae. Some uceptble populato uppoed to avod moquto bte ad ome fected hot are olated o that they do ot tramt the deae. A ytem of dfferetal equato that model the populato dyamc ad the aocated bac reproducto umber are dcued the paper. Stablty aaly made to determe the dyamcal behavor of the ytem. Keyword: Degue; Stablty; Awaree; Bac reproducto umber I. INTRODUCTION Degue fever a fectou vector bore deae preadg tropcal ad ubtropcal coutre. Degue tramtted by aede moqutoe. Four erotype of the degue vrue DEN 1, DEN, DEN 3 ad DEN 4 caue the degue fever. Nowaday, degue fever edemc more tha hudred coutre. I recet year, the umber of degue cae ha bee creag dramatcally. Awaree toward the deae ca chage the whole dyamc of the tramo of the deae. Due to awaree, people ca take dfferet kd of precauto toward the deae. Avodg moquto bte the major precauto agat degue fever. Some of the precauto that ca be take are: to keep home, evromet ad urroudg hygee, to remove all tagat water ad cotaer, to cover all cotaer properly to prevet degue moquto breedg there, to wrap all uued platc tyre, to ue moquto repellet to avod moquto bte, to ue aerool ad moquto col to kll moqutoe, to wear log leeve ad fully covered clothe, to ue moqutoe et aroud bed whle leepg etc. To cotrol the deae effectvely, oe hould udertad the dyamc of the deae tramo ad take all of the correpodg detal to accout. Kermack ad McKedrc cotrbuted o the developmet of the mathematcal theory of epdemc [1]. I the paper, the author codered three compartmet- Suceptble, Ifectou ad Removal for mathematcal formulato of the model. Eteva ad Varga made a tudy o the tramo of degue fever wth cotat huma populato [] ad varable vector populato [3]. Edy ad Suprata developed a tramo model for degue fever retrctg the dyamc for the cotat hot ad vector populato, ad reducg the model to two-dmeoal plaar equato [4]. Dfferet tude have bee made to vetgate the degue deae tramo [5 9]. The preet paper coder SIR model wth a fracto of 015, IJAERA - All Rght Reerved 39

2 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato uceptble hot ad fected hot populato aware of degue deae tramo. It uppoed that the populato do ot teract wth the moqutoe. II. FORMULATION OF THE MODEL To tudy tramo proce of the degue fever, hot populato dvded to three compartmet uceptble, fected ad recovered. People who are healthy ad may potetally get fected wth degue vru are codered to be uceptble compartmet, people who are fected wth degue ad are able to tramt the deae are codered to be fected compartmet ad people who have recovered from degue deae are codered to be recovered compartmet. The populato of moqutoe dvded to two compartmet oly, uceptble ad fected compartmet: moqutoe that may potetally become fected wth degue vru are codered to be uceptble compartmet ad moqutoe that are fected wth degue ad ca tramt the deae are codered to be fected compartmet. The recovered cla the moquto populato doe ot ext a ther fecto perod ed wth ther death. Table 1. Decrpto of tate varable h : Cotat hot (huma) populato ze h : Number of uceptble hot populato h : Number of fectve hot populato h : Number of removal (recovered) hot populato r m : Vector (moquto) populato ze m : Number of uceptble vector populato m : Number of fectve the vector populato I the preet model, the fracto p of uceptble hot populato aware of the deae tramo who ue moquto repellet, moqutoe et etc. to avod the moquto bte due to awaree codered. So, the populato doe ot come cotact wth the fected moqutoe. Alo, a fracto q of fected hot olated a a reult of awaree. Thu, the preet paper clude the effect of awaree the model propoed by Eteva ad Varga []. p q Table. The parameter ued the model ad ther dmeo : Fracto of uceptble hot populato aware of degue tramo, Dmeole : Fracto of fected hot populato aware of degue tramo, Dmeole h : Brth/death rate the hot populato, Tme -1 m : Death rate the vector populato, Tme -1 h m : Tramo probablty from vector to hot, Dmeole : Tramo probablty from hot to vector, Dmeole h : Recovery rate the hot populato, Tme -1 b : Btg rate of vector, Tme -1 A : Recrutmet rate, Moqutoe Tme , IJAERA - All Rght Reerved 40

3 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato The ytem of dfferetal equato whch decrbe the preet model are; For huma populato: dh h (1 p) hm hh dt dh (1 p) hm ( h h) h (1) dt dhr hh hhr dt For vector populato: dm mb A (1 q) mh mm dt () dm mb (1 q) mh mm dt Alo, h h hr A m m m (3) Above equato ca be reduced to three equato dh h (1 p) hm hh dt dh (1 p) hm ( h h) h (4) dt dm (1 q) m b A m h mm dt h m Itroducg the proporto h h m x, y, z (5) A / m We obta, dx h (1 x) xz dt dy xz y (6) dt dz (1 z) y z dt 015, IJAERA - All Rght Reerved 41

4 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato bh A where, (1 p), h h, (1 q) bm, m mh III. EQUILIBRIUM POINTS AND STABILITY ANALYSIS A. Deae free equlbrum pot I a deae free tuato, y 0, z 0. Hece, from (6), x 1. So, the deae free equlbrum pot (1, 0, 0). B. Bac Reproducto Number The bac reproducto umber the expected umber of ecodary fecto produced by a dex cae a completely uceptble populato. Accordg to the value of the bac reproducto umber, deae ca pert wth R0 1 ad the deae ca de out whe R 0 1. Ug the ext geerato method, we compute the bac reproducto umber R 0, aocated wth the deae free equlbrum (1, 0, 0). Ug lat two equato of the ytem of Equato (6), the o-egatve matrx F, of the fecto term ad the o-gular matrx V, of the trato term are gve, repectvely [10] by: Ad, therefore F 0 FV 0 ad V Egevalue of 1 FV are Therefore, the bac reproducto umber, R 0 = pectral radu of the matrx, Propoto 1 (Stablty of deae free equlbrum pot) 1 FV =. The deae free equlbrum pot aymptotcally table f R0 1 ad utable f R0 1. Proof: Jacoba matrx of ytem of equato (6) at the deae free equlbrum pot (1, 0, 0) h 0 J , IJAERA - All Rght Reerved 4

5 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato Ug the matrx J, we fd the followg charactertc equato: ( h )[ ( ) (1 R0 )] 0 (7) From charactertc equato (7), t oberved that frt egevalue h whch egatve. Next, ( ) (1 R0 ) 0 (8) The two codto of Routh Hurwtz Crtera, for local aymptotcal tablty of ecod order charactertc polyomal a1 a 0 are. a1 0. a 0 We have, a1 h h m whch alway potve. Alo, a (1 R0 ). Here, potve. Ad 1R0 0 f R0 1. So, a (1 R0 ) 0 f R0 1. Hece, f R0 1the egevalue wll have egatve real part ad the deae free equlbrum pot become aymptotcally table. If R0 1, the two egevalue of equato (8) are oe egatve ad oe potve real umber. So, the deae free equlbrum pot become utable f R0 1. Propoto (Extece of the Edemc equlbrum pot) The Edemc equlbrum pot of the ytem of equato (6) ext f R0 1. e e e Proof: Solvg the ytem of equato (6), the equlbrum pot foud are (1, 0, 0) ad ( x, y, z ) h ( ) ( ) e e e h h where ( x, y, z ),, ( h ) ( h ) ( h ) The frt pot deae free equlbrum pot. The ecod pot become edemc equlbrum pot ad ext f 0. Whch mple that 1 ad the R0 1. Hece, the edemc equlbrum pot ext f R0 1. IV. NUMERICAL RESULTS AND DISCUSSION I the preet paper we explored the effect of awaree the tramo of degue deae takg dfferet value of awaree parameter p ad q. For the value p = 0, q = 0 preet model chage to the model propoed by Eteva ad Varga []. To llutrate the dyamc of the degue deae, the dfferet value of the awaree fracto p ad q ued are gve Table 3. Table 3. Parameter ad ther value ued Parameter h m h m h b h A pq, value /day /day /day 1/ Varable referece [] [7] [6] [6] [] [7] [] [] --- Fgure 1 draw for p = 0 = q (o awaree). Fgure ad 3 (wth awaree) are draw for p = 0.5, q = 0.7 ad p = 0.8, q = 0.9 ad compared wth the Fgure 1. Fgure ad 3 llutrate the effect of awaree the tramo of the degue epdemc howg that there mall umber of people 015, IJAERA - All Rght Reerved 43

6 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato fected of deae ad large umber of people remaed uceptble whe compared wth Fgure 1. The ze of fected moquto populato ee maller Fgure ad 3 tha Fgure 1. Fgure 1. Dyamc of hot ad vector populato wth p = 0 = q. Fgure. Dyamc of hot ad vector populato wth p = 0.50, q = The fecto rate oberved hgher whe there o awaree (Fgure 1) tha whe there awaree people (Fgure ad Fgure 3). Alo, the rate of fecto ee decreag wth the creag value of awaree parameter (Fgure ad Fgure 3). Fgure 3 how that whe there uffcet umber of people aware of the deae tramo, there very le umber of hot ad vector fected of the deae. Thu, creag awaree people ca decreae the fecto rate ad coequetly ca decreae the umber of fected populato. Fgure 3. Dyamc of hot ad vector populato wth p = 0.80, q = Fgure 4. Bac reproducto umber wth dfferet value of awaree term p ad q. For p = 0 = q, the value of bac reproducto umber, R0 3.7 ad for p = 0.8, q = 0.9, the value of bac reproducto umber, R Thu, the value of bac reproducto umber le whe there o awaree ad more the preece of awaree hot populato. Alo, Fgure 4 how that the value of bac reproducto umber approache uty ad become le tha uty for uffcetly hgher value of awaree parameter p ad q. V. CONCLUSION A degue model wa tuded by cludg the mpact of the awaree parameter the tramo dyamc of degue fever. A aaly of the deae tramo wa made wth ad 015, IJAERA - All Rght Reerved 44

7 Iteratoal Joural of Advaced Egeerg Reearch ad Applcato wthout the preece of awaree the hot populato. A fracto of uceptble hot populato uppoed ot to come cotact wth moqutoe. The populato uppoed to be aware of the deae tramo ad they are uppoed to take all poble precauto uch a ug moquto repellet, moquto et etc o that they ca avod the bte of moqutoe. Alo, a fracto of fected hot populato uppoed ot to tramt the deae a the populato olated due to awaree. Small value of bac reproducto umber were appeared the preece of hgher level of awaree ad large value of bac reproducto umber wa appeared the abece of awaree hot populato. Hece, the pread of the deae come uder cotrol wth the creae awaree hot populato. For hgher level of awaree, the deae ee to affect le umber of people ad moqutoe. Large umber of people ee to be affected from the deae whe there o awaree the hot populato. So, the preet tudy ugget that wth the creae the awaree hot populato, remarkable uceptble hot populato ze ca be aved from beg fected. So, pread of awaree about the deae tramo play a mportat role cotrollg the degue deae tramo. VI. REFERENCES [1] W. O. Kermack ad A. G. McKedrck (197). A cotrbuto to the mathematcal theory of epdemc, Proceedg of the Royal Socety of Lodo, vol. 115, pp [] L. Eteva ad C. Varga (1998). Aaly of a degue deae tramo model, Math. Bo-cece, vol. 150, pp [3] L. Eteva ad C. Varga (1999). A model for degue deae wth varable huma populato, J. Math. Bol., vol. 38, pp [4] E. Soewoo ad A. K. Suprata (001). A two dmeoal model for the tramo of degue fever deae, Bull. Malaya Math. Sc. Soc., vol. 4, pp [5] P. Pogumpu (008). Mathematcal model of degue deae wth the cubato perod of vru, World Academy of Sc. Egg. ad Tech., vol. 44, pp [6] S. T. R. Pho, C. P. Ferrera, L. Eteva, F. R. Barreto, V. C. Morato e Slva ad M. G. L. Texera (010). Modellg the dyamc of degue real epdemc, Phl Tra. R. Soc., vol. 368, pp [7] R. Koguy ad P. Pogumpu (011). Mathematcal Modelg for Degue Tramo wth the Effect of Seao, Iteratoal joural of Mathematcal, Computatoal, Phycal, Electrcal ad Computer Egeerg, vol. 5, pp [8] S. Sde ad S. M. Noora (013). A SIR model for pread of degue fever deae (mulato for outh Sulawe, Idoea ad Selagor, Malaya, world Joural of Modellg ad Smulato, vol. 9, pp [9] S. Gakkhar ad N. C. Chavda (013). Impact of awaree o the pread of degue fecto huma populato, Appled Mathematc, vol. 4, pp [10] O. Dekma, J. A. P. Heeterbeek ad J. A. J. Metz (1990). O the defto ad computato of the bac reproducto rato R 0 model for fectou deae heterogeeou populato, Joural of Mathematcal Bology, vol. 8, pp , IJAERA - All Rght Reerved 45