Stability Analysis of the Euler Discretization for SIR Epidemic Model


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1 Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we cosider a discrete SIR epidemic model obtaied by the Euler method For that discrete model, existece of disease free equilibrium ad edemic equilibrium is established Sufficiet coditios o the local asymptotical stability of both disease free equilibrium ad edemic equilibrium are also derived It is foud that the local asymptotical stability of the existig equilibrium is achieved oly for a small time step size h If h is further icreased ad passes the critical value, the both equilibriums will lose their stability Our umerical simulatios show that a complex dyamical behavior such as bifurcatio or chaos pheomeo will appear for relatively large h Both aalytical ad umerical results show that the discrete SIR model has a richer dyamical behavior tha its cotiuous couterpart Keywords: Discrete SIR epidemic model, Euler method, Stability aalysis, Bifurcatio diagram PACS: 87Ed, 6Lj, 545a INTRODUCTION Mathematical models that describe dyamical behavior of ifectious diseases trasmissios have bee extesively studied [] Such mathematical modelsare usually writte assystem of oliear differetial equatiosoe of basic epidemic model divides the total populatio ito three classes: susceptible idividuals(s), ifective idividuals (I)ad idividuals who have recovered or who have the disease but o loger ifective (R) By assumig that the recovered idividuals will acquire lifelog immuity,we obtai the wellkow SIR epidemic model: ds S SI di SI I dr I R, where the positive timeivariat parameters,,, ad are the recruitmet (birth ad immigratio) rate, the atural death rate, the trasmissio disease coefficiet, the rate of recovery from disease ad the rate of death iduced by the disease, respectivelythe dyamics of model () are well uderstood The basic reproductio umber for system () is System () has always a disease free equilibrium (DFE) poit /,, E Furthermore, if the system () also has a edemic equilibrium (EE) poit E Se, Ie, Re, where S, e Ie ad R e The DFE poit is asymptotically stable if, while the EE poit is asymptotically stable wheever For practical purposes, for example due to the eed of scietific computatio ad realtimesimulatio, the cotiuous model has to be solved umerically Oe of the commo method for solvig a system of oliear () Proceedigs of the 3rd Iteratioal Coferece o Mathematical Scieces AIP Cof Proc 6, (4); doi: 63/ AIP Publishig LLC /$3 375 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Wed, 8 Oct 4 8::
2 differetial equatios is Euler method [35, 7] or ostadard fiite differece method [6, 8]I this paper, the cotiuous model () will be discretized usig theforward Euler scheme to get whereh is the time step size, f t I h S S h h h I S R hi h h R, I f ad t h,,,, We focus o the behaviors ofdiscrete dyamical system () The discrete system () is cosidered i [3] but oly for the case of uit time step, ie whe h = For this case, they derived sufficiet coditios for the global stability of the existig equilibrium The case of arbitrary time step h has bee studied i [4,5], but oly for the case of Here we study the dyamics of discrete system () for arbitrary time step had Specifically, we aim to derive the coditio for the local stabilityof the existig equilibrium We are also iterested i the dyamics of system () whe a equilibrium losses its stability () EQUILIBRIUM POINT AND ITS STABILITY Equilibrium Poits Equilibrium poits of discrete system () ca be foud by substitutig S S S ; I I I ; ad R R R ito () ad solvig the resultig system of equatios We fid that there always exists a DFE poit E /,,Furthermore, if the we also have a EE poit E Se, Ie, Re Therefore the equilibrium poits of discrete system () are cosistet with those of the cotiuous model () I the followig we study the local stability of each equilibrium poit by cosiderig the liearizatio of system () at the DFE poit ad E S, I, R is EE poit, respectively The Jacobia matrix of the liearized system at equilibrium J h hi hs h E hi hs h (3) Equilibrium E is asymptotically stable if the modulus of all eigevalues of the correspodig Jacobia matrix J E are less tha oe Stability of Disease Free Equilibrium The characteristic equatio of the Jacobia matrix (3) at the DFE poit E /,, The eigevalues of E h h (4) h ad J are h 3 h / It is clear that if h / It ca also be show that 3 if ad h / Hece, we have the followig theorem is 376 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Wed, 8 Oct 4 8::
3 Theorem The disease free equilibrium /,, E is asymptotically stable if ad h mi, At the edemic equilibrium poit E S I, R Stability of Edemic Equilibrium e, e e, the Jacobia matrix (3) has the followig characteristic equatio h A B (5) where h B h h Clearly that oe of the eigevalues of the Jacobia matrix is h while other eigevalues, 3are determied by A B if h / The modulus of other eigevalues will be studied usig the followig lemma A ad Lemma [6, ] The quadratic equatio A B has two roots that satisfy i, i, 3 if ad oly if the followig coditios are satisfied: (i) B, (ii) A B, (iii) A B From equatio (5), it ca be show that coditio B ad A B are respectively equivalet to h ad The coditio A B is the same as h h 4 By deotig D 4 6 we cosider three cases as follow: Case i: Suppose that D, the A B is satisfied if h Case ii: Suppose that D, the A B holds if Case iii: Suppose that D, the A B holds if h h D or D h Based o the above discussio, we obtai the followig theorem Theorem 3 The edemic equilibrium poit S I, R coditios holds: E is asymptotically stable if ad oe of the followig e, e e 377 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Wed, 8 Oct 4 8::
4 () D ad () D ad h mi, mi,, D h We remark that the stability aalysis of discrete model () with has bee discussed i [4] Here, the author used differet method whe ivestigatig the properties of the eigevalues of Jacobia matrix Nevertheless, for the case of, the coditios for both DFE poit ad EE poit to be asymptotically stable give by Theorem ad Theorem are i accordace with the coditios give i [4] NUMERICAL SIMULATIONS FIGURE Susceptible idividuals as fuctio of time calculated usig scheme () for differet values of h To illustrate the behavior of discrete SIR epidemic model (), we perform some umerical simulatios usig 445,, 5, 5, ad varyig the value of time step h Usig those parameters, we have 5669, E 5,,, 4,58676,4669 previous aalysis, the EE poit 4, 58676,4669 E ad D Accordig to the E is asymptotically stable if h 987 I Figure, we show our umerical results usig h 9, 9,, ad iitial value S () 39, I 5 7 This plot shows the umber of susceptible idividuals as fuctio of time It is show that if we take h 9, the the solutio is coverget to the EE poit as expected However, if we use h 987 the the EE poit loses its stability I particular, if we choose h 9,, the the umerical solutio shows that there exists periodic, 4periodic ad 8periodic solutio, respectively Takig higher value of h leads to solutio with chaotic behavior Such a behavior ca be see from the bifurcatio diagram as show i Figure Chaos occurs i the bads where the dotsseem to be smeared at radom This diagram shows that the discrete model () has richer dyamics compared to the origial cotiuous model () 378 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Wed, 8 Oct 4 8::
5 FIGURE Bifurcatio diagram CONCLUDING REMARKS We have aalyzed a discrete SIR epidemic model obtaied from the Euler scheme It is show that the stability of both DFE poit ad EE poit of the discrete system is cosistet with its origial cotiuous model oly if a relatively small time step (h) is used For relatively large time step, both equilibrium poits may lose their stability ad may show complex dyamical behavior Mathematically, such behavior is iterestig but it is ot dyamically cosistet with its related cotiuous model Therefore, we suggest usig other umerical methods whe solvig the SIR system, for example ostadard fiite differece method ACKNOWLEDGMENTS This research is fiacially supported by the Directorate Geeral of Higher Educatio, Miistry of Educatio ad Culture of Republic Idoesia via DIPA of Brawijaya Uiversity No:DIPA /3, ad based o letter of decisio of Brawijaya Uiversity Rector No: 95/SK/3 REFERENCES F Brauer ad C CastilloChavez, Mathematical models i populatio biology ad epidemiology, New York: Spriger Verlag, HW Hethcote, SIAM Rev 4, () 3 X Ma, Y Zhou ad H Cao, Advaces i Differece Equatios 3, 4 (3) 4 W Piyawog, SpatioTemporal Numerical Modellig of Whoopig Cough Dyamics, PhD Thesis, Bruel Uiversity, 5 Z Hu, Z Teg ad H Jiag, Noliear Aalysis: Real World Applicatios 3(5), 7 33 () 6 MY Ogu ad I Turha, Joural of Applied Mathematics 3, 9 (3) 7 T Fayeldi, A Suryato ad A Widodo, It J Appl Math Stat 47 (7), (3) 8 RE Mickes, Applicatio of ostadard fiite differece schemes, Sigapore: World Scietific Publishig Co Pte Ltd, 9 RE Mickes, Nostadard Fiite Differece Models of Differetial Equatios, Sigapore: World Scietific, 994 A Suryato, It J MathComput 3 (D), 3 () A Suryato, WM Kusumawiahyu, I Darti ad I Yati, Comput Appl Math 3(), (3) S Elaydi, A itroductio to differece equatios, New York: Spriger, This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Wed, 8 Oct 4 8::
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