Bifurcation of an Epidemic Model with Sub-optimal Immunity and Saturated Recovery Rate

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1 Bifuration of an Epidemi odel with Sub-optimal Immunit and Saturated Reoer Rate Chang Phang Department of Siene and athematis Fault of Siene Tehnolog and Human Deelopment Uniersiti Tun Hussein Onn alasia Yong Hong Wu Department of athematis and Statistis Curtin Uniersit Perth Australia Abstrat In this paper we stu the bifuration of an epidemi model with sub-optimal immunit and saturated treatment/reoer rate Different from lassial models suboptimal models are more realisti to eplain the miroparasite infetions disease suh as Pertussis and Influenza A B arring out the bifuration analsis of the model we show that for ertain alues of the model parameters Hopf bifuration Bogdono-Takens bifuration and its assoiated homolini bifuration our B stuing the bifuration ures we an predit the persistene or etintion of diseases Kewords- sub-optimal immunit; saturated treatment/reoer rate; Hopf bifuration; homolini bifuration; Bogdono-Takens bifuration; I INTRODUCTION In reent ears etensie researh has been arried out worldwide to deelop more realisti epidemi models For ompartmental ODE models seeral new models for inidene rate and treatment/reoer rate hae been introdued Subsequent analtial studies show that some of these epidemi models possess rih namis In a seminal paperuan and Wang [] presented a SIR epidemi model with the nonlinear inidene rate in the form of β SI /( In the paper the onsider a redued sstem and perform an elaboratie analsis of equilibrium through a quadrati equation Using transformation to normal form the show that the model undergoes Hopf bifuration homolini bifuration and Bogdono-Takens bifuration Following the paper a few other papers disuss about the same namial behaior in the SIR model but with different forms of inidene rates suh as β SI /( bi [] and β SI /( bi [] Similarl different treatment/reoer/remoal rate are onsidered in order to predit the trend of disease transmission more auratel Unlike the earlier model the reent models ma hae two endemi equilibria when R 0 < Hene the eradiation of disease depends not onl on R 0 but also on the initial sizes of all sub-populations The work in [] is a pioneer work for bifuration analsis whih shows the eistene of Hopf bifuration and Bogdono-Takens bifuration for the model with onstant remoal rate After the work of [] arious studies of bifuration for the models with other treatment/reoer rate hae been arried out Bakward bifuration is shown in a SIR model with pieewise funtion treatment in [] meanwhile the work in [6] laims the eistene of Hopf bifuration in a SIR model with saturated treatment rate Furthermore in the SIR model with saturated inidene rate and saturated treatment rate [7] onl bakward bifuration is shown to eist while referene [] suggests that a SIS model with saturated reoer rate possesses Bogdono-Takens bifuration Howeer to date no analsis has been done to stu the eistene of Bogdono-Takens bifuration in the SIR model with saturated reoer rate Hene we intend to further stu the bifuration of the SIR model and we will use the more generalized form of the model namel the sub-optimal immunit model whih lies in between the SIS and SIR models In this paper we undertake the bifuration analsis for an epidemi model with sub-optimal immunit and saturated treatment/reoer rate Apart from using the saturated treatment/reoer rate an additional parameter σ is used to form the sub-optimal immunit model as in [9] The new model lies in between the SIS and SIR models The suboptimal immunit model will be more appropriate for the stu of miroparasite infetions whih usuall ours during hildhood After a primar infetion one ma get temporar immunit (immune protetion will wane oer time or partial immunit (immunit that ma not full protetie Eamples of this kind of diseases inlude Pertussis (temporar immunit and Influenza (partial immunit [0] Different to that in [9] we show in this paper that Bogdono-Takens bifuration and its assoiated homolini bifuration eist in this sub-optimal immunit model Throughout the paper for sake of simpliit we hoose some speifi alues for the parameters as [] did The parameter alues an be easil replaed b other alues as long as the onditions are fulfilled Our analsis was arried out for the ase where the basi reprodution number R 0 is less than unit Apart from the disussion of Hopf bifuration we show that the sub-optimal immunit model undergoes Bogdono-Takens bifuration and its assoiated homolini bifuration II QUALITATIVE ANALYSIS We onsider a model with sub-optimal immunit and saturated reoer rate 0 IEEE International Conferene on Sstems Biolog (ISB //$600 0 IEEE Zhuhai China September 0

2 ( ds A βsi σt ( I µ S di βsi T ( I µ I dr ( σ T ( I µ R where all the parameters are positie A is the reruitment rate of suseptible population β is the disease transmission rate µ is the natural death rate and T (I is the reoer rate In our analsis at equilibrium point we assume that A I S I R and we take T ( I I in whih µ ( and are respetiel the reoer rate of the infeted population with and with no treatment b R ( Defining the basi reprodution number βa R 0 µ ( µ T (0 βa 0 µ ( µ Define with R I T ( I I we obtain βaa βk( βµ Now we onsider the following redued sstem di A I ( I R I I µ I β µ dr I k( I µ R k σ At equilibrium di 0 0 and hene from ( we obtain R ki ( ( µ ( dr Then b substituting this into the first equation in (after some algebra work we obtain [ β a ( k I [ β ( k k ]I µ ( µ ν βa 0 ( ( Let [ ] β ( k k [ βa( k [ µ ( µ ν βa] Lemma a Sstem ( has a unique positie equilibrium E ( I under an of the following three onditions i R and [ ( k Aa a( 0 β k µ µ µ [ β ( ] whih I βa( µ ii R 0 > for whih [ β ( ] I [ βa( iii 0 ki ( ( µ ( ki ( ( µ ( and [ ( k Aa a( β k µ µ µ [ β ( ] whih I [ βa( ki ( ( µ ( for for b Sstem ( has two positie equilibria E ( I R and E I if and onl if ( R R 0 < > 0 where β k k µ Aa µ a µ and [ ( ( ] < 0 [ β ( ] I [ βa( [ β ( ] I [ βa( The Jaobian matri for sstem ( is A β I β ( µ I R k ( The determinant of is as follows ( β a ( k µ ( det( I ki ( ( µ ( ki ( ( µ ( ( µ ( β a ( k k µ Aa µ a ( I ( β ( µ Aa k k µ a( µ I µ ( µ ν βa βi µ The sign of the determinant is determined b the sign of β a ( k S µ I ( β a ( k k µ Aa µ a ( I µ ( β ( µ Aa k k µ a( µ I µ ( µ ν βa Using ( we get ( β a( Aa k k µ µ a ( I S µ ( β ( µ k k µ a I µ ( µ ν βa Lemma a The unique positie equilibrium E ( I in sstem ( is i a degenerate equilibrium if 0 [ ( k k β ii a enter-tpe equilibrium if R 0 > while tr ( 0 b The positie equilibrium E ( I R in sstem ( leads to S ( I < 0 while > 0 0 < and [ β ( k k It is thus a saddle point The positie equilibrium E ( I R in sstem ( leads to S ( I 0 while > 0 and > 0 < 0 IEEE International Conferene on Sstems Biolog (ISB //$600 0 IEEE 6 Zhuhai China September 0

3 [ β ( k k or enter It is thus a node fous (9 d u u u u u O( III HOPF BIFURCATION In this setion we will show that the model in ( undergoes Hopf bifuration for some alues Let ( β a µ k ( for I and set ( R tr ( 0 then we obtain A while I ( 0 < This happens when R ( Replaing I and R b and namel ( I R ( we hae ( ( 9 To translate to the origin we set X ( Y and rename XY as respetiel Then (6 (7 ( ( ( ( ( 9( ( ( ( ( ( Using the Talor epansion for (6 we hae 9 7 ( O( O( The Jaobian matri for (7 at is 7 ( > 6 We thus hae tr ( 0 and det( A 0 and Hopf bifuration ours B arring out transformation X Y and then renaming XY as respetiel (7 beomes ( O( O( 6 aking the hange of ariables obtain u we du u u u u O( (0 6 Let k and F ( u u u u O( F ( u u u u O( We an get the first Liapuno onstant σ b σ 6 6 k F F 66 9 F F F F F F F F F F F F Hene there is an unstable periodi orbit when A inreases from In the following we hoose A as a bifuration parameter Let A ε From ( we obtain ( ( 9 ε It is eas to show that ε 6ε 6ε 0 ε 9 6ε 6ε ( is the positie equilibrium of the sstem ( The Jaobian matri is gien b where ε 6 ( ( Hene the harateristi equation is gien b ( 0 m ± m mc A B We thus obtain where m A m B 9 67ε 66ε 6ε ( 9 7ε 6ε 6ε 6ε ε 6ε 0 IEEE International Conferene on Sstems Biolog (ISB //$600 0 IEEE 7 Zhuhai China September 0

4 m C 0676ε 99ε 9960ε 9ε ( ε ε i Re ( ε 0 when ε 0 ii Im ( ε 0 when ε iii Re d ( ε 0 when ε 0 dε Theorem There eist a σ > 0 and a funtion ε ε ( defined on 0 < σ whih satisf ε ( 0 and when ε ε ( < 0 sstem ( has a unique unstable limit le whih passes through ( Fig shows an unstable orbit for sstem ( when ( β a µ k ( and A ε 0ε ε 6ε 6ε To translate ( to the origin we set X Y and rename XY as respetiel Then 7(0 6ε 6ε (7 6ε 6ε 6ε ( ( 9 ( ( ( ( ( Hene we hae 6 ( ( ( ( ( ( ( Using the Talor epansion for ( we hae ( O( O( The Jaobian matri for ( at ( is We thus hae tr ( 0 and det( A 0 Clearl the matri has two zero eigenalues and thus the Bogdano- Takens bifuration ours B arring out transformation X Y and renaming XY as respetiel ( beomes O( Figure An unstable periodi orbit when ( β a µ k ( and A IV BOGDANOV-TAKENS BIFURCATION In this setion we will stu the Bogdano-Takens bifuration for some alues of the model in ( We hoose ( β a µ k ( for ( I and let R 0 and we obtain Setting A 9 we obtain ( I ( trae ( 0 and det( 0 Writing I and R as and namel ( I ( we hae ( 9 ( O( In order to obtain the anonial normal form we follow the proedure as in [] Setting u we obtain (6 du O( d u u O( In the following we find the uniersal unfolding of ( I ( b hoosing parameters A and as bifuration parameters in a small neighbourbood of ( β a µ k ( Let A 9 and We hae (7 ( ( 9 ( ( 0 IEEE International Conferene on Sstems Biolog (ISB //$600 0 IEEE Zhuhai China September 0

5 To translate ( to the origin we set X Y and rename XY as respetiel Then ( ( 9 ( ( ( ( ( ( ( ( ( ( ( ( ( ( Using the Talor epansion for ( we hae Let 6 ( ( ( O( ( 6 ( 6 0 ( ( O( (9 X 6 Y ( 6 6 ( ( O( and rename XY as respetiel Then we obtain a 0 a a a a a O( 7 6 where a 0 a a a and B setting as we hae where a X a a a (ie X and rewriting X b0 b a a a O( a 0 6 b b a and a B rewriting the equation using the new time τ with a dτ (ie ( dτ and then rewriting τ ( as t we obtain 6 where b and 0 a O( 0 6 b a B the hange of ariables a a 7 6 X Y τ t and then renaming X Yτ as t respetiel we obtain τ τ O( where τ 0 andτ b a τ a B putting τ and simplifing it sstem ( has a saddle-node bifuration and the saddle-node bifuration ure is gien b Theorem At the Bogdano point the model ( with ( β a µ k ( A 9 and in a small neighbourhood of ( I ( bifuration : a has the following i saddle-node bifuration: the saddle-node bifuration ure is gien b b O ( 0 ii Hopf bifuration : the Hopf bifuration ure is gien 6 0 O ( 0 iii Homolini bifuration : the homolini bifuration ure is gien b O ( 0 Fig shows the homolini bifuration when for sstem (7 ( ( b0 b a a a O( ( Carring out the transformation X Y ( and then renaming XY as respetiel we hae 0 IEEE International Conferene on Sstems Biolog (ISB //$600 0 IEEE 9 Zhuhai China September 0

6 Figure Homolini bifuration when From the result in Theorem we stu the bifuration ures near the origin on the ( plane The ures pass through the origin and there are four regions separated b these bifuration ures If we take near 0 we obtain the region as in Figure The Jaobian matri for sstem (7 is where ( ( ( ( and ( 0 ( E beomes an unstable fous and the limit le disappears In this stage at finite time an positie orbits eept for the two equilibria E and E will tend to the ais R 0 ie the disease beomes etint When ( lies in region IV there is no positie equilibrium and the disease will disappear The lassifiation of the equilibrium points an be easil heked b the eigenalues of the Jaobian matri V CONCLUSION In this paper we hae proposed an epidemi model with sub-optimal immunit and saturated treatment/reoer rate Through global analsis the sstem in ( has been shown to hae rih namial behaiour inluding Hopf bifuration Bogdono-Takens bifuration and its assoiated homolini bifuration We also show that when the bifuration parameters are within ertain regions the disease will be persistent or etint ACKNOWLEDGENT The first author would like to thank Uniersiti Tun Hussein Onn alasia for supporting his PhD stu REFERENCES Figure The four tpial regions separated b the bifuration ures The horizontal ais is the -ais and the ertial ais is the -ais If we take 0 after some simple alulation we obtain the result as shown in the Table below TABLE I THE CLASSIFICATION OF EQUILIBRIU POINTS E i det( tr( Q Conlusion I 090 E (- ( ( Unstable saddle E ( (- (- Stable fous II 096 E (- ( ( Unstable saddle E ( (- (- Stable fous III 060 E (- ( ( Unstable saddle E ( ( (- Unstable fous IV 060 No positie equilibrium Q tr( det( ( ( When ( lies in region I as in Figure there is no limit le or homolini orbit and E is a stable fous If ( lies in region II there is a unique limit le inside the positie orbits of sstem (7 and the orbits approah E as t tends to infinit In this situation the disease is persistent inside the le When lies in region III ( [] S G Ruan and W Wang Dnamial behaiour of an epidemi model with a nonlinear inidene rate J Differential Equations ol 00 pp-6 [] Y G Zho D Xiao and Y L Li Bifuration of an epidemi model with non-monotoni inidene rate of saturated mass ation Chaos Solitons and Fratals ol 007 pp90-9 [] Z G Song J Xu and Q H Li Loal and global bifuration in an SIR epidemi model Applied athematis and Computation ol 009 pp-7 [] W Wang and S G Ruan Bifuration in an epidemi model with onstant remoal rate of the infeties J ath Anal Appl ol 9 00 pp77-79 [] W Wang Bakward bifuration of an epidemi model with treatment athematial Biosiene ol pp-7 [6] Z H Zhang and Y H Suo Qualitatie analsis of a SIR epidemi model with saturated treatment rate J Appl ath Comput ol 00 pp77-9 [7] X Zhang and X N Li Bakward bifuration of an epidemi model with saturated treatment funtion J ath Anal Appl ol 00 pp- [] J G Cui X X u and H Wan Saturation reoer leads to multiple endemi equilibria and bakward bifuration Journal of Theoretial Biolog ol 00 pp7- [9] J H Pang J A Cui and J Hui Rih namis of an epidemi model with sub-optimal immunit and nonlinear reoer rate athematial and Computer odelling in press [0] G Gomes L J White and GF edle Infetion reinfetion and aination under suboptimal immune protetion: epidemiologial perspeties Journal of Theoretial Biolog ol 00 pp9-9 [] G J Peng and Y L Jiang Pratial omputation of normal forms of the Bogdono-Takens bifuration Nonlinear Dn Springer Siene 0 0 IEEE International Conferene on Sstems Biolog (ISB //$600 0 IEEE 60 Zhuhai China September 0

Linear and Nonlinear State-Feedback Controls of HOPF Bifurcation

IOSR Journal of Mathematis (IOSR-JM) e-issn: 78-578, p-issn: 19-765X. Volume 1, Issue Ver. III (Mar. - Apr. 016), PP 4-46 www.iosrjournals.org Linear and Nonlinear State-Feedbak Controls of HOPF Bifuration